Session Y33: Decoherence and Quantum Control

Single spin decoherence by general spin chains

Spin decoherence induced by a spin bath has recently been the subject of interest in the field of quantum computation and spintronics. Unlike the spin-boson model, the resulting decoherence depends crucially on the nature of the spin bath and its coupling to the central spin. In this work we investigate the decoherence of a central spin which is coupled non-uniformly to a spin chain by means of the time-dependent density matrix renormalization group technique. Using this technique the coupling between the central spin and the spin chain can take any form, in contrast to the typical uniform or on-site coupling taken in the literature. We have studied the resulting spin decoherence induced by spin chains in the Ising, XY, XXZ, and Heisenberg universality classes. Connection between the decoherence the quantum phase transition of the spin chain is discussed.   

Dynamical decoupling induced renormalization of the non-Markovian dynamics

In this work we develop a numerical framework to investigate the renormalization of the non-Markovian dynamics of an open quantum system to which dynamical decoupling is applied. We utilize a non-Markovian master equation which is derived from the non-Markovian quantum trajectories formalism. It contains incoherent Markovian dynamics and coherent Schr\"odinger dynamics as its limiting cases and is capable of capture the transition between them. We have performed comprehensive simulations for the cases in which the system is either driven by the Ornstein-Uhlenbeck noise or or is described by the spin-boson model. The renormalized dynamics under bang-bang control and continuous dynamical decoupling are simulated. Our results indicate that the renormalization of the non- Markovian dynamics depends crucially on the spectral density of the environment and the envelop of the decoupling pulses. The framework developed in this work hence provides an unified approach to investigate the efficiency of realistic decoupling pulses. This work also opens a way to further optimize the decoupling via pulse shaping.   

Coherence oscillations produced by non-Gaussian quantum noise

The usual models for dissipative environments involve a bath of harmonic oscillators, producing Gaussian fluctuations. However, modern experiments on dephasing in qubits and electronic interferometers indicate strong coupling to non-Gaussian quantum noise. Most strikingly, the coherence (interference contrast) may oscillate as a function of time and other control parameters. We present the theory behind a recent ``controlled dephasing'' experiment involving an electronic Mach-Zehnder interferometer strongly coupled to the non-Gaussian shot noise of a detector edge channel [cond-mat/0610634,cond-mat/ 0611372], as well as applications to qubits dephased by shot noise or two-level fluctuators.   

Optimal Control of Large Spin Systems

A quantum system is said to be controllable if the accessible Hamiltonians (as a Lie algebra) generate all unitary operators on Hilbert space. Optimal quantum state control seeks a time-dependent sequence of Hamiltonians that maximize the fidelity with an arbitrary target state given a fixed initial state. We consider optimal control of the spin of a cesium atom restricted to its F=3 ground state hyperfine manifold, with a Hilbert space of dimension 2F+1=7. Control is implemented through time varying magnetic fields in two orthogonal directions along with a quadratic AC-Stark shift created by an off-resonant laser probe. The optimization is performed under several constraints, most importantly a temporal limitation determined by dephasing due to photon scattering and parameter inhomogeneity. The fidelity of state preparation is verified through both a full density matrix simulation and reconstruction from experimental data.   

The effect of qubit-qubit exchange interaction on qubit relaxation rates

In this report we extend an exactly soluble model of one-qubit decoherence to two qubits with exchange interactions. Closed-form expressions for the transfer matrix can be obtained, but contrary to the single-qubit case, the matrix must be diagonalized numerically. We compute the single- and two-qubit relaxation rates. In the first approximation, the two-qubit rates can be obtained by additivity of single-qubit rates. Interactions and the resulting entanglement modified this, but their effect is surprisingly small. These results suggest that N-qubit decoherence rates scale linearly with N.   

Resilient Quantum Computation in Correlated Environments: A Quantum Phase Transition Perspective

The `threshold theorem' is a central result in the theory of quantum error correction. It was derived initially for a stochastic error model, but relentless effort has been dedicated to including correlated errors. Here, we demonstrate that a large class of correlated error models is reduced to the simple stochastic model in the asymptotic limit of large number of qubits or long time. Thus, in order to prove the resilience of the quantum information for these models, we can fall back on the traditional derivation of the threshold theorem. Because the conditions for this fall back have clear parallels with the theory of quantum phase transitions, we rephrase the threshold theorem as a dimensional criterion: (1) For systems above their ``critical dimension'', the traditional proof of resilience is valid, and there are two regimes, or phases, as a function of the coupling with the environment. (2) However, when the system is below its ``critical dimension'', correlations produce large corrections, and it is not possible to prove resilience by our arguments.   

Linked cluster expansion of a qubit decoherence

We present a theoretical approach to study evolution of a qubit affected by a coupling with a spin bath. The procedure based on a linked cluster decomposition of system and bath dynamics. Unlike previous studies the approach allows exact evaluation of terms of each perturbative order in the exponent contributing to qubit decoherence and phase fluctuations. The procedure has a simple diagrammatic representation. We have utilized the theory to evaluate decoherence of a localized electron spin subject to an interaction with a nuclear spin bath. The novel results we report on are effects of nuclear spin clusters on electrons spin decoherence beyond pair correlation models and also control for dissipation processes in a spin environment.   

Collective decoherence of nuclear spin clusters

The problem of dipole–dipole decoherence of nuclear spins is considered for strongly entangled spin clusters. We consider the pure dephasing part of the dipole–dipole interaction which can be classically interpreted as a random field fluctuating along the quantization axes. Due to the long (but finite) range nature of dipole–dipole interaction this field is expected to be partially correlated at the sites of different spins in the cluster. Consequently our results show that the dynamics of the entangled spin cluster can be described as the decoherence due to interaction with a composite bath consisting of fully correlated and uncorrelated parts. The correlated term causes the slower decay of coherence at larger times. The decoherence rate scales up as a square root of the number of spins, giving the linear scaling of the resulting error. Our theory is consistent with recent experiments reported on decoherence of correlated spin clusters.   

Experimental demonstration of stimulated polarization wave in a chain of nuclear spins

A one-dimensional Ising chain irradiated by weak resonant transverse field is the simplest model of quantum amplifier [Phys.\ Rev.\ A 71, 062338 (2005)]. The quantum state of the chain is stationary when all the qubits (spins) are in the same state. However, when the first qubit is flipped, it triggers a stimulated wave of flipped qubits, propagating through the chain. Such ``quantum domino" dynamics induces huge change in the total polarization, a macroscopic observable. Here we present the experimental demonstration of this quantum amplification process on a four-qubit system by using nuclear magnetic resonance technique. The physical system is a linear chain of four $^{13}$C nuclear spins in a molecule of fully $^ {13}$C-labeled sodium butyrate dissolved in D$_2$O. The pseudopure ground state (with all spins up) is prepared by multi-frequency partial saturation. The wave of flipped spins has been clearly observed when the first spin of the chain is flipped. We define a coefficient of amplification as the relative enhancement of the total polarization change. In our experimental system, the measured coefficient of amplification is about 3.   

Initial quantum corrections, quantum process tomography, and physicality of not completely positive maps

We make the connection between initial quantum correlations and not completely positive maps. Though this has been suggested in literature for some time now, our arguments are supported by explicit calculations. In the process we will discuss our work in relation with quantum process tomography. We are especially interested in recent experiments that yielded so called non-physical results. We will offer new interpretations of these experiments and show that the results do make sense in the end.   

Zero Discord leads to Completely Positive Maps

The stochastic evolution of a quantum system can be expressed by a dynamical map that acts as a superoperator on a density matrix. If all eigenvalues of this map are positive, the map is said to be completely positive. If the dynamical map comes from the reduced unitary evolution of a bipartite system, the map depends on the correlations, and can have negative eigenvalues. Quantum discord is a measure of the quantumness of a correlation. A state with zero discord has the properties that the only correlations that it has are equivalent to the classical conditional probability. We prove that states with zero quantum discord always lead to completely positive maps. The connection with the proper preparation of states for experiments is made.   

Refocusing of a Qubit System Coupled to an Oscillator.

Within an NMR-like approach to coherent control, we analyze the performance of ``soft'' refocusing pulses and pulse sequences in protecting the coherence of a qubit system coupled to a quantum oscillator. We focus on the effects of the oscillator excitation and heating and associated deterioration of qubits' fidelity. These effects cannot be addressed in the conventional master equation formalism with the bath assumed in thermal equilibrium. Analytically, we construct the effective Hamiltonian of the controlled qubit plus oscillator system to quadratic order of the Magnus expansion in powers of the couplings. The qubit error operators and the terms responsible for the oscillator excitation are thus identified explicitly. These terms dominate the oscillator evolution when it is close to resonance with the qubit(s). The corresponding single- and few-qubit simulations show continuously increasing average oscillator energy accompanied by deteriorating qubit fidelity. The magnitude of the oscillator frequency bias needed to arrest this run-away effect is smaller for second-order refocusing sequences, where the order of the sequence is the number of suppressed terms in the effective Hamiltonian of the qubit system, with the oscillator operators replaced by $c$-numbers.   

Real-time dynamics of dissipative quantum systems

Dissipation in quantum systems is the source of a number of interesting and important phenomena. In quantum computing, for instance, environmental memory can have a significant effect on the operation of solid state devices and on error correction. In condensed matter, strong dissipation can cause phase transitions, as in the ubiquitous spin-boson model. In an effort to create a generic computational method for studying real-time non-Markovian and strongly dissipative dynamics, we have examined the construction of master equations containing memory. We have found this approach lacking because most of the physics beyond weak coupling is contained within the memory kernel of the master equation. Therefore, the majority of the effort in solving for the dynamics goes into the calculation of a system specific memory kernel. We discuss these issues as well as a potential solution based on the use of ancillary systems which represent part of the environment.