Time-dependent Hilbert spaces, geometric phases, and general covariance in quantum mechanics

We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schrödinger time-evolution identifies the metric with a positive-definite (Ermakov–Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that changes the metric of the Hilbert space but leaves the Hamiltonian of the system invariant. We propose to identify this phenomenon with a quantum mechanical analogue of the principle of general covariance of general relativity. We comment on the implications of this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.     

Experimental Hamiltonian Learning of an 11-Qubit Solid-State Quantum Spin Register

Learning the Hamiltonian of a quantum system is indispensable for prediction of the system dynamics and realization of high fidelity quantum gates.However,it is a significant challenge to efficiently characterize the Hamiltonian which has a Hilbert space dimension exponentially growing with the system size.Here,we develop and implement an adaptive method to learn the effective Hamiltonian of an 11-qubit quantum system consisting of one electron spin and ten nuclear spins associated with a single nitrogen-vacancy center in a diamond.We validate the estimated Hamiltonian by designing universal quantum gates based on the learnt Hamiltonian and implementing these gates in the experiment.Our experimental result demonstrates a well-characterized 11-qubit quantum spin register with the ability to test quantum algorithms,and shows our Hamiltonian learning method as a useful tool for characterizing the Hamiltonian of the nodes in a quantum network with solid-state spin qubits.     

Quantum Mechanical Virial Theorem in Systems with Translational and Rotational Symmetry

Generalized virial theorem for quantum mechanical nonrelativistic and relativistic systems with translational and rotational symmetry is derived in the form of the commutator between the generator of dilations G and the Hamiltonian H. If the conditions of translational and rotational symmetry together with the additional conditions of the theorem are satisfied, the matrix elements of the commutator [G,H] are equal to zero on the subspace of the Hilbert space. Normalized simultaneous eigenvectors of the particular set of commuting operators which contains H, J ², J _z and additional operators form an orthonormal basis in this subspace. It is expected that the theorem is relevant for a large number of quantum mechanical N-particle systems with translational and rotational symmetry.     

An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere

For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time τ. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.     

The quantum correlation dynamics of two qubits in finite-temperature environments with dynamical decoupling pulses

We investigate the dynamics of quantum correlation between two noninteracting qubits each inserted in its own finite-temperature environment with 1/f$1/f$ spectral density. It is found that the phenomenon of sudden transition between classical and quantum decoherence exists in the system when two qubits are initially prepared in X-type quantum states, and the transition time depends on the initial-state of two qubits, the qubit–environment coupling constant and the temperature of the environment. Furthermore, we explore the influence of dynamical decoupling pulses on the transition time and show that it can be prolonged by applying the dynamical decoupling pulses.     

On the theory of collective motion in nuclei. II. Quantum theory

The collective Hamiltonian is assumed to be invariant under the orthogonal group O(A ? 1, $\text{R}$). It is shown that the quantum collective dynamics can be formulated on a coset space of the symplectic group $\text{sp}$(6, $\text{R}$) of dimension 12, 16 or 18. The first case corresponds to the collective dynamics based on closed shells and leads to a Hilbert space of analytic functions in six complex collective quasiparticle variables. Dequantization of this scheme yields the classical dynamics described in I. In the limit A ? 1 one obtains a system of s- and d-bosons with symmetry group $\text{u}$ (6) in the collective state space.     

Vacuum structures in Hamiltonian light-front dynamics

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front Hamiltonian must annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure.     

Controllable coupling and quantum correlation dynamics of two double quantum dots coupled via a transmission line resonator

We propose a theoretical scheme to generate a controllable and switchable coupling between two double-quantum-dot (DQD) spin qubits by using a transmission line resonator (TLR) as a bus system. We study dynamical behaviors of quantum correlations described by entanglement correlation (EC) and discord correlation (DC) between two DQD spin qubits when the two spin qubits and the TLR are initially prepared in X-type quantum states and a coherent state, respectively. We demonstrate that in the EC death regions there exist DC stationary states in which the stable DC amplification or degradation can be generated during the dynamical evolution. It is shown that these DC stationary states can be controlled by initial-state parameters, the coupling, and detuning between qubits and the TLR. We reveal the full synchronization and anti-synchronization phenomena in the EC and DC time evolution, and show that the EC and DC synchronization and anti-synchronization depends on the initial-state parameters of the two DQD spin qubits. It is shown that the initial quantum correlation may be suppressed completely when the evolution time approaches to the infinity in the presence of dissipation. These results shed new light on dynamics of quantum correlations.     

Engineering massive quantum memories by topologically time-modulated spin rings

We introduce a general scheme to realize perfect storage of quantum information in systems of interacting qubits. This novel approach is based on global external controls of the Hamiltonian that yield time-periodic inversions in the dynamical evolution allowing a perfect periodic quantum state reconstruction. We illustrate the method in the particularly interesting and simple case of spin systems affected by XY residual interactions with or without static imperfections. The global control is achieved by step time inversions of an overall topological phase of the Aharonov-Bohm type. Such a scheme holds both at finite size and in the thermodynamic limit, thus enabling the massive storage of arbitrarily large numbers of local states, and is stable against several realistic sources of noise and imperfections.     

Quantum Darwinism in a Composite System: Objectivity versus Classicality

We investigate the implications of quantum Darwinism in a composite quantum system with interacting constituents exhibiting a decoherence-free subspace. We consider a two-qubit system coupled to an N-qubit environment via a dephasing interaction. For excitation preserving interactions between the system qubits, an analytical expression for the dynamics is obtained. It demonstrates that part of the system Hilbert space redundantly proliferates its information to the environment, while the remaining subspace is decoupled and preserves clear non-classical signatures. For measurements performed on the system, we establish that a non-zero quantum discord is shared between the composite system and the environment, thus violating the conditions of strong Darwinism. However, due to the asymmetry of quantum discord, the information shared with the environment is completely classical for measurements performed on the environment. Our results imply a dichotomy between objectivity and classicality that emerges when considering composite systems.     

Temperature dependence of bound state spectra in field theory

We analyze the behaviour of kinks and semiclassical bound states at finite temperatures by applying quantum statistics to the fluctuations which determine the quantum dynamics of these states. We consider two theories in one space dimension — the ?⁴ theory with a dynamical symmetry breaking and the Gross-Neveu model. For the ?⁴ theory, the one-loop temperature corrections are obtained by using temperature-dependent Green function techniques. We show that the same result can be obtained by applying quantum statistics to the fluctuations around the kink. For the Gross-Neveu model, the temperature dependence of the bound states, which correspond to time-independent field configurations, is obtained. We show that for every bound state there exists a critical temperature at which this state breaks up into its constituents. This critical temperature increases with the number of constituents of the bound state.     

Non-Classical Correlations and Transfer of Quantum Information in a Superconducting Qubit System with Dynamical Decoupling Pulses

We investigate the dynamics of non-classical correlations(entanglement and quantum discord) of the system consisting of two non-interacting superconducting qubits coupling with a common data bus, where the system is driven by the dynamical decoupling pulses. It is found that the non-classical correlations between two superconducting qubits can be increased by appling a train of dynamical decoupling pulses. Furthermore, we also explore the influence of the dynamical decoupling pulses on the information flowing between superconducting qubits and data bus by making use of the trace distance. It is shown that the dynamical decoupling pulses can protect quantum information of two superconducting qubits and force information to flow back to the superconducting qubits from the data bus.

     

Laplace transform approach for the dynamics of N qubits coupled to a resonator

An approach to use the method of Laplace transform for the perturbative solution of the Schrödinger equation at any order of the perturbation for a system of N qubits coupled to a cavity with n photons is suggested. We investigate the dynamics of a system of N superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations.     

Evaluation of an algebraic model for the vibrations of water,effects of a discrete symmetry

We evaluate the dynamics of an algebraic model Hamiltonian for the vibrational motion of the water molecule. We pay special attention to the effects of the discrete symmetry of order 2 of the model. For a comparison between the quantum dynamics and the classical dynamics it is necessary to desymmetrize such quantum states which are based on types of motion which come in symmetry related pairs. For the other states based on motion invariant under the symmetry operation a desymmetrization would be meaningless. The desymmetrized quantum states show a simple connection to the guiding motions of the classical dynamics which can be used for a complete assignment of the states even though the system is not integrable in the sense of Liouville and shows chaotic behaviour in large parts of the classical phase space.     

The su q(2) algebra in the off-diagonal basis and applications to quantum optics

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su _q(2) algebra. The corresponding matrix elements of finite rotations (the q-deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su _q(2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.     

Thermal Entanglement Between Two Two-Level Atoms in a Two-Photon Jaynes-Cummings Model with an Added Kerr Medium

In this paper, we consider a Hamiltonian model that includes interaction of two coupled two-level atoms with a single-mode quantized electromagnetic field in a cavity via the degenerate two-photon transition. The cavity is filled with a Kerr-like medium and is held at a temperature T. The free field Hamiltonian possesses the su(1,1) symmetry which realized by either even or odd photon-number states. The total number of excitation as a constant of motion, provides a decomposition of the Hilbert space of system into direct sums of invariant subspaces. As a results, the representation of the Hamiltonian becomes block-diagonal matrix with three blocks. After diagonalizing each block, we obtain thermal state of system in the whole Hilbert space and within its excitation subspaces. Finally, the effect of temperature, atom-atom and Kerr-type couplings on the degree of thermal entanglement between the atoms are investigated. Our results show that within the single-excitation subspace spanned with odd photon-number states, the entanglement between the atoms is thermally robust.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

D = 1 Supergravity and Quantum Cosmology

Using a D = 1 supergravity framework I construct a super-Friedmann equation for an isotropic and homogenous universe including dynamical scalar fields. In the context of quantum theory this becomes an equation for a wave function of the universe of spinorial type, the Wheeler–DeWitt–Dirac equation. It is argued that a cosmological constant breaks a certain chiral symmetry of this equation, a symmetry in the Hilbert space of universe states, which could protect a small cosmological constant from being affected by large quantum corrections.

     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.