Showing 1–33 of 33 results for author: Fan, W

Minimal nonintegrable models with three-site interactions

Authors: Wen-Ming Fan, Kun Hao, Yang-Yang Chen, Xiao-Hui Wang, Kun Zhang, Vladimir Korepin

Abstract: We study integrability breaking in translationally invariant spin-$1/2$ chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the… ▽ More We study integrability breaking in translationally invariant spin-$1/2$ chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the simplest models needed for a rigorous nonintegrability test, from the Hamiltonian density of the deformed Fredkin spin chain. Among the sixteen such models, six satisfy the Reshetikhin condition and are integrable, while two satisfy Hokkyo's criterion and are nonintegrable away from special coefficient values. Returning to one-site translation-invariant spin-$1/2$ chains leaves four genuine three-site models: two integrable models and two generically nonintegrable models. These results give a compact classification of minimal three-site models obtained from the deformed Fredkin spin chain and identify a local mechanism by which coupling individually integrable structures can destroy integrability beyond nearest-neighbor interactions. △ Less

Submitted 20 August, 2026; v1 submitted 8 February, 2026; originally announced February 2026.

Comments: 18 pages, 3 figures, 6 tables

Generalized Time-Coarse Graining via an Operator Cumulant Expansion

Authors: Leon Bello, Tal Rubin, Wentao Fan, Nathaniel Fisch, Hakan Türeci

Abstract: We introduce a general framework for deriving effective dynamics from arbitrary time-dependent generators, based on a systematic operator cumulant expansion. Unlike traditional approaches, which typically assume periodic or adiabatic driving, our method applies to systems with general time dependencies and is compatible with any dynamics generated by a linear operator -- Hamiltonian or not, quantu… ▽ More We introduce a general framework for deriving effective dynamics from arbitrary time-dependent generators, based on a systematic operator cumulant expansion. Unlike traditional approaches, which typically assume periodic or adiabatic driving, our method applies to systems with general time dependencies and is compatible with any dynamics generated by a linear operator -- Hamiltonian or not, quantum or classical, open or closed. This enables modeling of systems exhibiting strong modulation, dissipation, or non-adiabatic effects. Our approach unifies Hamiltonian techniques such as Lie-transform Perturbation Theory (LPT) with averaging-based methods like Time-Coarse Graining (TCG), revealing their structural equivalence through the lens of generalized cumulants. It also clarifies how non-Hamiltonian terms naturally emerge from averaging procedures, even in closed systems. We illustrate the power and flexibility of the method by analyzing a damped, parametrically driven Kapitza pendulum, a system beyond the reach of standard tools, demonstrating how accurate effective equations can be derived across a wide range of regimes. △ Less

Submitted 30 September, 2025; originally announced October 2025.

Absence of local conserved charges of the Fredkin spin chain and its truncated versions

Authors: Wen-Ming Fan, Kun Hao, Yang-Yang Chen, Kun Zhang, Xiao-Hui Wang, Vladimir Korepin

Abstract: Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstr… ▽ More Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare. △ Less

Submitted 23 November, 2025; v1 submitted 5 September, 2025; originally announced September 2025.

Comments: Close to published version, 20 pages, 16 tables

Journal ref: Phys. Rev. B 112, 205124 (2025)

Accelerating BP-based decoders for QLDPC Codes with Local Syndrome-Based Preprocessing

Authors: Wenxuan Fan, Yasunari Suzuki, Gokul Subramanian Ravi, Yosuke Ueno, Ilkwon Byun, Koji Inoue, Teruo Tanimoto

Abstract: Due to the high error rate of qubits, detecting and correcting errors is essential for achieving fault-tolerant quantum computing (FTQC). Quantum low-density parity-check (QLDPC) codes are one of the most promising quantum error correction (QEC) methods due to their high encoding rates. BP (Belief Propagation)-based decoders are widely used and highly competitive for QLDPC codes because BP offers… ▽ More Due to the high error rate of qubits, detecting and correcting errors is essential for achieving fault-tolerant quantum computing (FTQC). Quantum low-density parity-check (QLDPC) codes are one of the most promising quantum error correction (QEC) methods due to their high encoding rates. BP (Belief Propagation)-based decoders are widely used and highly competitive for QLDPC codes because BP offers inherent parallelism and strong scalability. However, BP-based decoders still suffer from high decoding latency, a large portion of which is spent in the iterative BP stage. In this paper, we propose a lightweight preprocessing step that utilizes local patterns in the syndrome to detect likely trivial error events and provide them as hints to BP-based decoders. These hints accelerate BP convergence and thereby reduce the overall decoding time. The proposed preprocessing step offers a broadly compatible approach to reducing the latency of BP-based QLDPC decodes. On the bivariate bicycle code $[[144,12,12]]$ at low physical error rates, our method achieves a $10\times$ speedup in decoding time for BP-OSD, and more than $2\times$ speedup for both BP-LSD and Relay-BP. Our method maintains the logical error rate when combined with BP-OSD and Relay-BP, while further achieving a significant reduction in logical error rate when combined with BP-LSD. △ Less

Submitted 17 April, 2026; v1 submitted 1 September, 2025; originally announced September 2025.

Long-time soliton dynamics via a coarse-grained space-time method

Authors: Dung N. Pham, Zoe Zager, Wentao Fan, Hakan E. Türeci

Abstract: We investigate the long-time dynamics of the Sine-Gordon (SG) model under a class of perturbations whose quantum field theoretic analog - via bosonization - corresponds to the massive Schwinger model describing 1+1D relativistic QED of Dirac fermions. Classical SG solutions offer critical insight into non-perturbative effects in this quantum theory, but capturing their long-time behavior poses sig… ▽ More We investigate the long-time dynamics of the Sine-Gordon (SG) model under a class of perturbations whose quantum field theoretic analog - via bosonization - corresponds to the massive Schwinger model describing 1+1D relativistic QED of Dirac fermions. Classical SG solutions offer critical insight into non-perturbative effects in this quantum theory, but capturing their long-time behavior poses significant numerical challenges. To address this, we extend a coarse-graining method to spacetime using a dual-mesh construction based on the Minkowski-metric. We first validate the approach against the well-studied variant of the SG model describing magnetic fluxon dynamics in Josephson transmission lines (JTLs), where analytical and numerical benchmarks exist. We then apply the method to the Schwinger-inspired SG model and uncover long-lived bound states - "Schwinger atoms" - in which a soliton is trapped by a fixed central charge. In certain regimes, the system exhibits limit cycles that give rise to positronium-like states of oppositely charged solitons, while in others such formation is suppressed. Accessing such long-time solutions requires a rigorous implementation of outgoing boundary conditions on a finite computational domain that provide radiative dissipation to allow relaxation toward states that exist only in an infinite domain. Here we provide such a construction. Our results also suggest the possibility of analog quantum simulation of relativistic quantum field theories with JTLs. These results demonstrate the utility of spatio-temporal coarse-graining methodology for probing non-perturbative structure formation in non-linear field theories. △ Less

Submitted 25 November, 2025; v1 submitted 16 April, 2025; originally announced April 2025.

Statistical features of quantum chaos using the Krylov operator complexity

Authors: Zhuoran Li, Wei Fan

Abstract: We study the statistical properties of Lanczos coefficients over an ensemble of random initial operators generating the Krylov space. We propose two statistical quantities that are important in characterizing the complexity: the average correlation matrix $\langle x_{i} x_{j}\rangle$ of Lanczos coefficients and the resulting distribution of the variance of Lanczos coefficients. Their resulting sta… ▽ More We study the statistical properties of Lanczos coefficients over an ensemble of random initial operators generating the Krylov space. We propose two statistical quantities that are important in characterizing the complexity: the average correlation matrix $\langle x_{i} x_{j}\rangle$ of Lanczos coefficients and the resulting distribution of the variance of Lanczos coefficients. Their resulting statistics are the Wishart distribution and the (rescaled) chi-square distribution respectively, which are independent of the distributions of initial operators and become the normal distribution in the case of large matrix size. As a numerical example, we use the typical billiard system with an integrability-breaking term and choose samples of random initial operators from given probability distributions (GOE, GUE and the uniform distribution). It agrees with the phenomenological analysis and further interesting behaviors are obtained, which indicates a consistent connection between RMT, Anderson localization and Krylov complexity. △ Less

Submitted 19 March, 2025; v1 submitted 27 November, 2024; originally announced November 2024.

Comments: In V3: the Stadium billiard is added. In the writing sytle, the phenomenology and the numerics are separated

Model Order Reduction for Open Quantum Systems Based on Measurement-adapted Time-coarse Graining

Authors: Wentao Fan, Hakan E. Türeci

Abstract: Model order reduction encompasses mathematical techniques aimed at reducing the complexity of mathematical models in simulations while retaining the essential characteristics and behaviors of the original model. This is particularly useful in the context of large-scale dynamical systems, which can be computationally expensive to analyze and simulate. Here, we present a model order reduction techni… ▽ More Model order reduction encompasses mathematical techniques aimed at reducing the complexity of mathematical models in simulations while retaining the essential characteristics and behaviors of the original model. This is particularly useful in the context of large-scale dynamical systems, which can be computationally expensive to analyze and simulate. Here, we present a model order reduction technique to reduce the time complexity of open quantum systems, grounded in the principle of measurement-adapted coarse-graining. This method, governed by a coarse-graining time scale $τ$ and the spectral band center $ω_0$, organizes corrections to the lowest-order model which aligns with the RWA Hamiltonian in certain limits, and rigorously justifies the resulting effective quantum master equation (EQME). The focus on calculating to a high degree of accuracy only what can be resolved by the measurement introduces a principled regularization procedure to address singularities and generates low-stiffness models suitable for efficient long-time integration. Furthermore, the availability of the analytical form of the EQME parameters significantly boosts the interpretive capabilities of the method. As a demonstration, we derive the fourth-order EQME for a challenging problem related to the dynamics of a superconducting qubit under high-power dispersive readout in the presence of a continuum of dissipative waveguide modes. This derivation shows that the lowest-order terms align with previous results, while higher-order corrections suggest new phenomena. △ Less

Submitted 20 April, 2026; v1 submitted 30 October, 2024; originally announced October 2024.

Journal ref: Phys. Rev. X 16, 021015, 2026

Systematic time-coarse graining for driven quantum systems

Authors: Leon Bello, Wentao Fan, Aditya Gandotra, Hakan E. Türeci

Abstract: Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relie… ▽ More Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relies on effective Hamiltonian models that introduce additional nonlinear processes which approximate the long-time dynamics so that highly oscillatory terms may be ignored. However, the removal of such high frequency transitions can only be performed rigorously within a systematic framework of time-coarse graining, which is a fundamentally irreversible operation. This implies that standard approaches with unitary effective models cannot accurately capture the long-time behavior of strongly driven nonlinear quantum systems in general. We introduce a systematic perturbation theory for obtaining the complete non-unitary effective model of the time-coarse grained (TCG) dynamics of a driven quantum system to any order in the coupling strengths. We derive a closed-form analytical formula for both unitary and non-unitary contributions, in the form of an effective Hamiltonian and non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory presumes unitary time evolution at the microscopic level, the time-coarse grained dynamics is found to follow a non-unitary time evolution in general. This occurs even when there is no open heat reservoir for the system to become entangled with or dissipate into. We demonstrate the effectiveness of the new method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for. △ Less

Submitted 31 July, 2024; v1 submitted 8 July, 2024; originally announced July 2024.

Multiplexed entanglement swapping with atomic-ensemble-based quantum memories in the single excitation regime

Authors: Minjie Wang, Haole Jiao, Jiajin Lu, Wenxin Fan, Shujing Li, Hai Wang

Abstract: Entanglement swapping (ES) between memory repeater links is critical for establishing quantum networks via quantum repeaters. So far, ES with atomic-ensemble-based memories has not been achieved. Here, we experimentally demonstrated ES between two entangled pairs of spin-wave memories via Duan-Lukin-Cirac-Zoller scheme. With a cloud of cold atoms inserted in a cavity, we produce non-classically-co… ▽ More Entanglement swapping (ES) between memory repeater links is critical for establishing quantum networks via quantum repeaters. So far, ES with atomic-ensemble-based memories has not been achieved. Here, we experimentally demonstrated ES between two entangled pairs of spin-wave memories via Duan-Lukin-Cirac-Zoller scheme. With a cloud of cold atoms inserted in a cavity, we produce non-classically-correlated spin-wave-photon pairs in 12 spatial modes and then prepare two entangled pairs of spin-wave memories via a multiplexed scheme. Via single-photon Bell measurement on retrieved fields from two memories, we project the two remaining memories never entangled previously into an entangled state with the measured concurrence of C = 0.0124(0.003). The successful probability of ES in our scheme is increased by three times, compared with that in non-multiplexed scheme. Our presented work shows that the generation of entanglement (C>0) between the remaining memory ensembles requires the average cross-correlation function of the spin-wave-photon pairs to be >30 . △ Less

Submitted 31 December, 2023; originally announced January 2024.

Unify the effect of anharmonicity in double-wells and anharmonic oscillators

Authors: Wei Fan, Huipen Zhang, Zhuoran Li

Abstract: We study the effect of anharmonicity in quantum anharmonic oscillators, by computing the energy gap between the ground and the 1st excited state using the numerical bootstrap method. Based on perturbative formulae of limiting coupling regimes, we propose a qualitative formula of the energy gap across all coupling values. Except detailed numerical parameter values, the proposed formula has the same… ▽ More We study the effect of anharmonicity in quantum anharmonic oscillators, by computing the energy gap between the ground and the 1st excited state using the numerical bootstrap method. Based on perturbative formulae of limiting coupling regimes, we propose a qualitative formula of the energy gap across all coupling values. Except detailed numerical parameter values, the proposed formula has the same functional form as the formula of ground state level splitting of double-well potentials, proposed recently in arXiv:2308.11516. This unifies the effects of anharmonicity in both the anharmonic oscillators and the double-well potentials, although the underlying physical process of them are completely different. We give an explanation of this connection of their anharmonicity from the viewpoint of quantum phase transitions. This connection is justified up to the octic anharmonicities by the numerical bootstrap. △ Less

Submitted 12 June, 2024; v1 submitted 17 September, 2023; originally announced September 2023.

Comments: 15 pages, 5 figures. arXiv admin note: text overlap with arXiv:2308.11516; In V2, add fig6

Proof-of-principle demonstration of temporally multiplexed quantum repeater link based on atomic ensemble

Authors: Minjie Wang, Haole Jiao, Jiajin Lu, Wenxin Fan, Zhifang Yang, Mengqi Xi, Shujing Li, Hai Wang

Abstract: Duan-Lukin-Cirac-Zoller quantum repeater protocol provides a feasible scheme to implement long-distance quantum communication and large-scale quantum networks. The elementary link, namely the entanglement between two atomic ensembles, is a fundamental component of quantum repeater. For practical quantum repeater, it is required that the elementary link can be prepared with high yield and the spin… ▽ More Duan-Lukin-Cirac-Zoller quantum repeater protocol provides a feasible scheme to implement long-distance quantum communication and large-scale quantum networks. The elementary link, namely the entanglement between two atomic ensembles, is a fundamental component of quantum repeater. For practical quantum repeater, it is required that the elementary link can be prepared with high yield and the spin waves stored in atoms can be efficiently converted into photons on demand. However, so far, such quantum repeater link has not been demonstrated in experiments. Here, we demonstrate a proof-of-principle multiplexed quantum repeater link by entangling two temporally multiplexed quantum memory. Compared with a single-mode link, the successful preparation rate of the multiplexed link is increased by one order of magnitude. By using the cavity-enhanced scheme, the on-demand retrieval efficiency of atomic spin waves is improved to 70%, which is beneficial for the subsequent entanglement swapping between adjacent links. The realization of temporally multiplexed quantum repeater link with high retrieval efficiency lays a foundation for the development of practical quantum networks. △ Less

Submitted 28 August, 2023; originally announced August 2023.

Cavity-enhanced and spatial-multimode spin-wave-photon quantum interface

Authors: Minjie Wang, Haole Jiao, Jiajin Lu, Wenxin Fan, Zhifang Yang, Mengqi Xi, Shujing Li, Hai Wang

Abstract: Practical realizations of quantum repeaters require quantum memory simultaneously providing high retrieval efficiency, long lifetime and multimode storages. So far, the combination of high retrieval efficiency and spatially multiplexed storages into a single memory remains challenging. Here, we set up a ring cavity that supports an array including 6 TEM00 modes and then demonstrated cavity enhance… ▽ More Practical realizations of quantum repeaters require quantum memory simultaneously providing high retrieval efficiency, long lifetime and multimode storages. So far, the combination of high retrieval efficiency and spatially multiplexed storages into a single memory remains challenging. Here, we set up a ring cavity that supports an array including 6 TEM00 modes and then demonstrated cavity enhanced and spatially multiplexed spin wave photon quantum interface (QI). The cavity arrangement is according to Fermat' optical theorem, which enables the six modes to experience the same optical length per round trip. Each mode includesn horizontal and vertical polarizations. Via DLCZ process in a cold atomic ensemble, we create non classically correlated pairs of spin waves and Stokes photons in the 12 modes. The retrieved fields from the multiplexed SWs are enhanced by the cavity and the average intrinsic retrieval efficiency reaches 70% at zero delay. The storage time for the case that cross-correlation function of the multiplexed QI is beyond 2 reaches 0.6ms . △ Less

Submitted 24 July, 2023; originally announced July 2023.

Fisher information as general metrics of quantum synchronization

Authors: Yuan Shen, Hong Yi Soh, Leong-Chuan Kwek, Weijun Fan

Abstract: Quantum synchronization has emerged as a crucial phenomenon in quantum nonlinear dynamics with potential applications in quantum information processing. Multiple measures for quantifying quantum synchronization exist. However, there is currently no widely agreed metric that is universally adopted. In this paper, we propose using classical and quantum Fisher information (FI) as alternative metrics… ▽ More Quantum synchronization has emerged as a crucial phenomenon in quantum nonlinear dynamics with potential applications in quantum information processing. Multiple measures for quantifying quantum synchronization exist. However, there is currently no widely agreed metric that is universally adopted. In this paper, we propose using classical and quantum Fisher information (FI) as alternative metrics to detect and measure quantum synchronization. We establish the connection between FI and quantum synchronization, demonstrating that both classical and quantum FI can be deployed as more general indicators of quantum phase synchronization, in some regimes where all other existing measures fail to provide reliable results. We show advantages in FI-based measures, especially in 2-to-1 synchronization. Furthermore, we analyze the impact of noise on the synchronization measures, revealing the robustness and susceptibility of each method in the presence of dissipation and decoherence. Our results open up new avenues for understanding and exploiting quantum synchronization. △ Less

Submitted 12 June, 2023; originally announced June 2023.

Comments: 9 pages, 12 figures

Journal ref: Entropy 2023, 25(8), 1116

Enhancing quantum synchronization through homodyne measurement, noise and squeezing

Authors: Yuan Shen, Hong Yi Soh, Weijun Fan, Leong-Chuan Kwek

Abstract: Quantum synchronization has been a central topic in quantum nonlinear dynamics. Despite rapid development in this field, very few have studied how to efficiently boost synchronization. Homodyne measurement emerges as one of the successful candidates for this task, but preferably in the semi-classical regime. In our work, we focus on the phase synchronization of a harmonic-driven quantum Stuart-Lan… ▽ More Quantum synchronization has been a central topic in quantum nonlinear dynamics. Despite rapid development in this field, very few have studied how to efficiently boost synchronization. Homodyne measurement emerges as one of the successful candidates for this task, but preferably in the semi-classical regime. In our work, we focus on the phase synchronization of a harmonic-driven quantum Stuart-Landau oscillator, and show that the enhancement induced by homodyne measurement persists into the quantum regime. Interestingly, optimal two-photon damping rates exist when the oscillator and driving are at resonance and with a small single-photon damping rate. We also report noise-induced enhancement in quantum synchronization when the single-photon damping rate is sufficiently large. Apart from these results, we discover that adding a squeezing Hamiltonian can further boost synchronization, especially in the semi-classical regime. Furthermore, the addition of squeezing causes the optimal two-photon pumping rates to shift and converge. △ Less

Submitted 18 July, 2023; v1 submitted 26 February, 2023; originally announced February 2023.

Comments: 6 pages, 8 figures

Journal ref: Phys. Rev. E 108, 024204, 2023

Quantum synchronization effects induced by strong nonlinearities

Authors: Yuan Shen, Wai-Keong Mok, Changsuk Noh, Ai Qun Liu, Leong-Chuan Kwek, Weijun Fan, Andy Chia

Abstract: A paradigm for quantum synchronization is the quantum analog of the Stuart--Landau oscillator, which corresponds to a van der Pol oscillator in the limit of weak (i.e. vanishingly small) nonlinearity. Due to this limitation, the quantum Stuart--Landau oscillator fails to capture interesting nonlinearity-induced phenomena such as relaxation oscillations. To overcome this deficiency we propose an al… ▽ More A paradigm for quantum synchronization is the quantum analog of the Stuart--Landau oscillator, which corresponds to a van der Pol oscillator in the limit of weak (i.e. vanishingly small) nonlinearity. Due to this limitation, the quantum Stuart--Landau oscillator fails to capture interesting nonlinearity-induced phenomena such as relaxation oscillations. To overcome this deficiency we propose an alternative model which approximates the van der Pol oscillator to finitely large nonlinearities while remaining numerically tractable. This allows us to uncover interesting phenomena in the deep-quantum strongly-nonlinear regime with no classical analog, such as the persistence of amplitude death on resonance. We also report nonlinearity-induced position correlations in reactively coupled quantum oscillators. Such coupled oscillations become more and more correlated with increasing nonlinearity before reaching some maximum. Again, this behavior is absent classically. We also show how strong nonlinearity can enlarge the synchronization bandwidth in both single and coupled oscillators. This effect can be harnessed to induce mutual synchronization between two oscillators initially in amplitude death. △ Less

Submitted 20 June, 2023; v1 submitted 7 January, 2023; originally announced January 2023.

Comments: 13 pages, 8 figures

Journal ref: Phys. Rev. A, 107.053713, (2023)

DEC-QED: A flux-based 3D electrodynamic modeling approach to superconducting circuits and materials

Authors: Dung N. Pham, Wentao Fan, Michael G. Scheer, Hakan E. Türeci

Abstract: Modeling the behavior of superconducting electronic circuits containing Josephson junctions is crucial for the design of superconducting information processors and devices. In this paper, we introduce DEC-QED, a computational approach for modeling the electrodynamics of superconducting electronic circuits containing Josephson junctions in arbitrary three-dimensional electromagnetic environments. D… ▽ More Modeling the behavior of superconducting electronic circuits containing Josephson junctions is crucial for the design of superconducting information processors and devices. In this paper, we introduce DEC-QED, a computational approach for modeling the electrodynamics of superconducting electronic circuits containing Josephson junctions in arbitrary three-dimensional electromagnetic environments. DEC-QED captures the non-linear response and induced currents in BCS superconductors and accurately captures phenomena such as the Meissner effect, flux quantization and Josephson effects. Using a spatial coarse-graining formulation based on Discrete Exterior Calculus (DEC), DEC-QED can accurately simulate transient and long-time dynamics in superconductors. The expression of the entire electrodynamic problem in terms of the gauge-invariant flux field and charges makes the resulting classical field theory suitable for second quantization. △ Less

Submitted 16 April, 2023; v1 submitted 24 December, 2022; originally announced December 2022.

Photoionization detection of a single Er$^{3+}$ ion with sub-100-ns time resolution

Authors: Yangbo Zhang, Wenda Fan, Jiliang Yang, Hao Guan, Qi Zhang, Xi Qin, Changkui Duan, Gabriele G. de Boo, Brett C. Johnson, Jeffrey C. McCallum, Matthew J. Sellars, Sven Rogge, Chunming Yin, Jiangfeng Du

Abstract: Efficient detection of single optical centers in solids is essential for quantum information processing, sensing, and single-photon generation applications. In this work, we use radio-frequency (RF) reflectometry to electrically detect the photoionization induced by a single Er$^{3+}$ ion in Si. The high bandwidth and sensitivity of the RF reflectometry provide sub-100-ns time resolution for the p… ▽ More Efficient detection of single optical centers in solids is essential for quantum information processing, sensing, and single-photon generation applications. In this work, we use radio-frequency (RF) reflectometry to electrically detect the photoionization induced by a single Er$^{3+}$ ion in Si. The high bandwidth and sensitivity of the RF reflectometry provide sub-100-ns time resolution for the photoionization detection. With this technique, the optically excited state lifetime of a single Er$^{3+}$ ion in a Si nano-transistor is measured for the first time to be 0.49 $\pm$ 0.04 $μ$s. Our results demonstrate an efficient approach for detecting a charge state change induced by Er excitation and relaxation. This approach could be used for fast readout of other single optical centers in solids and is attractive for large-scale integrated optical quantum systems thanks to the multi-channel RF reflectometry demonstrated with frequency multiplexing techniques. △ Less

Submitted 1 December, 2022; originally announced December 2022.

Multi-state Swap Test Algorithm

Authors: Wen Liu, Han-Wen Yin, Zhi-Rao Wang, Wen-Qin Fan

Abstract: Estimating the overlap between two states is an important task with several applications in quantum information. However, the typical swap test circuit can only measure a sole pair of quantum states at a time. In this study we designed a recursive quantum circuit to measure overlaps of multiple quantum states $|φ_1...φ_n\rangle$ concurrently with $O(n\log n)$ controlled-swap (CSWAP) gates and… ▽ More Estimating the overlap between two states is an important task with several applications in quantum information. However, the typical swap test circuit can only measure a sole pair of quantum states at a time. In this study we designed a recursive quantum circuit to measure overlaps of multiple quantum states $|φ_1...φ_n\rangle$ concurrently with $O(n\log n)$ controlled-swap (CSWAP) gates and $O(\log n)$ ancillary qubits. This circuit enables us to get all pairwise overlaps among input quantum states $|\langleφ_i|φ_j\rangle|^2$. Compared with existing schemes for measuring the overlap of multiple quantum states, our scheme provides higher precision and less consumption of ancillary qubits. In addition, we performed simulation experiments on IBM quantum cloud platform to verify the superiority of the scheme. △ Less

Submitted 14 May, 2022; originally announced May 2022.

The Zeeman and hyperfine interactions of a single $^{167}Er^{3+}$ ion in Si

Authors: Jiliang Yang, Wenda Fan, Yangbo Zhang, Changkui Duan, Gabriele G. de Boo, Rose L. Ahlefeldt, Jevon J. Longdell, Brett C. Johnson, Jeffrey C. McCallum, Matthew J. Sellars, Sven Rogge, Chunming Yin, Jiangfeng Du

Abstract: Er-doped Si is a promising candidate for quantum information applications due to its telecom wavelength optical transition and its compatibility with Si nanofabrication technologies. Recent spectroscopic studies based on photoluminescence excitation have shown multiple well-defined lattice sites that Er occupies in Si. Here we report the first measurement of the Zeeman and hyperfine tensors of a s… ▽ More Er-doped Si is a promising candidate for quantum information applications due to its telecom wavelength optical transition and its compatibility with Si nanofabrication technologies. Recent spectroscopic studies based on photoluminescence excitation have shown multiple well-defined lattice sites that Er occupies in Si. Here we report the first measurement of the Zeeman and hyperfine tensors of a single 167Er3+ ion in Si. All the obtained tensors are highly anisotropic with the largest value principal axes aligning in nearly the same direction, and the trace of the lowest crystal field level g-tensor is 17.78$\pm$0.40. The results indicate that this specific Er site is likely to be a distorted cubic site that exhibits monoclinic (C1) symmetry. Finally, zero first-order-Zeeman (ZEFOZ) fields are identified for this site and could be used to reduce decoherence of hyperfine spin states in future experiments. △ Less

Submitted 24 April, 2022; originally announced April 2022.

Journal ref: Phys. Rev. B 105, 235306 (2022)

Spectral broadening of a single Er$^{3+}$ ion in a Si nano-transistor

Authors: Jiliang Yang, Jian Wang, Wenda Fan, Yangbo Zhang, Changkui Duan, Guangchong Hu, Gabriele G. de Boo, Brett C. Johnson, Jeffrey C. McCallum, Sven Rogge, Chunming Yin, Jiangfeng Du

Abstract: Single rare-earth ions in solids show great potential for quantum applications, including single photon emission, quantum computing, and high-precision sensing. However, homogeneous linewidths observed for single rare-earth ions are orders of magnitude larger than the sub-kilohertz linewidths observed for ensembles in bulk crystals. The spectral broadening creates a significant challenge for achie… ▽ More Single rare-earth ions in solids show great potential for quantum applications, including single photon emission, quantum computing, and high-precision sensing. However, homogeneous linewidths observed for single rare-earth ions are orders of magnitude larger than the sub-kilohertz linewidths observed for ensembles in bulk crystals. The spectral broadening creates a significant challenge for achieving entanglement generation and qubit operation with single rare-earth ions, so it is critical to investigate the broadening mechanisms. We report a spectral broadening study on a single Er$^{3+}$ ion in a Si nano-transistor. The Er-induced photoionisation rate is found to be an appropriate quantity to represent the optical transition probability for spectroscopic studies, and the single ion spectra display a Lorentzian lineshape at all optical powers in use. Spectral broadening is observed at relatively high optical powers and is caused by spectral diffusion on a fast time scale. △ Less

Submitted 27 January, 2022; originally announced January 2022.

Journal ref: Phys. Rev. Applied 18, 034018 (2022)

GHZ-like states in the Qubit-Qudit Rabi Model

Authors: Yuan Shen, Giampiero Marchegiani, Gianluigi Catelani, Luigi Amico, Ai Qun Liu, Weijun Fan, Leong-Chuan Kwek

Abstract: We study a Rabi type Hamiltonian system in which a qubit and a d-level quantum system (qudit) are coupled through a common resonator. In the weak and strong coupling limits the spectrum is analysed through suitable perturbative schemes. The analysis show that the presence of the multilevels of the qudit effectively enhance the qubit-qudit interaction. The ground state of the strongly coupled syste… ▽ More We study a Rabi type Hamiltonian system in which a qubit and a d-level quantum system (qudit) are coupled through a common resonator. In the weak and strong coupling limits the spectrum is analysed through suitable perturbative schemes. The analysis show that the presence of the multilevels of the qudit effectively enhance the qubit-qudit interaction. The ground state of the strongly coupled system is a found of Greenberger-Horne-Zeilinger (GHZ) type. Therefore, despite the qubit-qudit strong coupling, the nature of the specific tripartite entanglement of the GHZ state suppress the bipartite entanglement. We analyze the system dynamics under quenching and adiabatic switching of the qubit-resonator and qudit-resonator couplings. In the quench case, we found that the non-adiabatic generations of photons in the resonator is enhanced by the number of levels in the qudit. The adiabatic control represents a possible route for preparation of GHZ states. Our analysis provides relevant information for future studies on coherent state transfer in qubit-qudit systems. △ Less

Submitted 5 July, 2021; v1 submitted 26 April, 2021; originally announced April 2021.

Comments: 15 pages, 7 figures, Submission to SciPost

Journal ref: SciPost Phys. 11, 099 (2021)

Effect of Size and Shape on Electronic and Optical Properties of CdSe Quantum Dots

Authors: Yincheng Liu, Sumanta Bose, Weijun Fan

Abstract: In this paper, we used the 8-band k$\cdot$p model with valence force field considerations to investigate the effect of size and shape on electronic and optical properties of cadmium selenide quantum dots. Major factors related to their properties including band mixing probabilities, spatial charge distributions, transition matrix elements and Fermi factors were studied. Volumetrically larger CdSe… ▽ More In this paper, we used the 8-band k$\cdot$p model with valence force field considerations to investigate the effect of size and shape on electronic and optical properties of cadmium selenide quantum dots. Major factors related to their properties including band mixing probabilities, spatial charge distributions, transition matrix elements and Fermi factors were studied. Volumetrically larger CdSe dots were found to have smaller band-gaps but higher transition matrix elements and Fermi factors. The maximum optical gain for dots was observed to have an initially positive and then negative correlation with their real-space size as a result of combined effects of various factors. For the shape effects, cubic dots were found to have smaller band-gaps, Fermi factors and transition matrix elements than spherical dots due to higher level of asymmetry and different surface effects. Consequently, cubic dots have lower emission energy, smaller amplification. The occurrence of near E1-H1 transition broadens the gain spectrum of cubic dots. Cubic and spherical dots are both proven to be promising candidates for optical devices under visible range. We have demonstrated that size and shape change could both effectively alter the properties of quantum dots and therefore recommend consideration of both when optimizing the performance for any desired application. △ Less

Submitted 5 November, 2017; originally announced November 2017.

Comments: Published in Optik - International Journal for Light and Electron Optics (8 pages, 10 figures), 2017

Optoelectronics of Inverted Type-I CdS/CdSe Core/Crown Quantum Ring

Authors: Sumanta Bose, Weijun Fan, Dao Hua Zhang

Abstract: Inverted type-I heterostructure core/crown quantum rings (QRs) are quantum-efficient luminophores, whose spectral characteristics are highly tunable. Here, we study the optoelectronic properties of type-I core/crown CdS/CdSe QRs in the zincblende phase - over contrasting lateral size and crown width. For this we inspect their strain profiles, transition energies, transition matrix elements, spatia… ▽ More Inverted type-I heterostructure core/crown quantum rings (QRs) are quantum-efficient luminophores, whose spectral characteristics are highly tunable. Here, we study the optoelectronic properties of type-I core/crown CdS/CdSe QRs in the zincblende phase - over contrasting lateral size and crown width. For this we inspect their strain profiles, transition energies, transition matrix elements, spatial charge densities, electronic bandstructure, band-mixing probabilities, optical gain spectra, maximum optical gains and differential optical gains. Our framework uses an effective-mass envelope function theory based on the 8-band k$\cdot$p method employing the valence force field model for calculating the atomic strain distributions. The gain calculations are based on the density-matrix equation and take into consideration the excitonic effects with intraband scattering. Variations in the QR lateral size and relative widths of core and crown (ergo the composition) affect their energy levels, band-mixing probabilities, optical transition matrix elements, emission wavelengths/intensity, etc. The optical gain of QRs is also strongly dimension and composition dependent with further dependency on the injection carrier density causing band-filling effect. They also affect the maximum and differential gain at varying dimensions and compositions. △ Less

Submitted 24 October, 2017; originally announced October 2017.

Comments: Published in AIP Journal of Applied Physics (11 pages, 7 figures)

Journal ref: Sumanta Bose, Weijun Fan, Dao Hua Zhang, Journal of Applied Physics 122, 163102 (2017)

Electronic bandstructure and optical gain of lattice matched III-V dilute nitride bismide quantum wells for 1.55 $μ$m optical communication systems

Authors: W. J. Fan, Sumanta Bose, D. H Zhang

Abstract: Dilute nitride bismide GaNBiAs is a potential semiconductor alloy for near- and mid-infrared applications, particularly in 1.55 $μ$m optical communication systems. Incorporating dilute amounts of Bismuth (Bi) into GaAs reduces the effective bandgap rapidly, while significantly increasing the spin-orbit-splitting energy. Additional incorporation of dilute amounts of Nitrogen (N) helps to attain lat… ▽ More Dilute nitride bismide GaNBiAs is a potential semiconductor alloy for near- and mid-infrared applications, particularly in 1.55 $μ$m optical communication systems. Incorporating dilute amounts of Bismuth (Bi) into GaAs reduces the effective bandgap rapidly, while significantly increasing the spin-orbit-splitting energy. Additional incorporation of dilute amounts of Nitrogen (N) helps to attain lattice matching with GaAs, while providing a route for flexible bandgap tuning. Here we present a study of the electronic bandstructure and optical gain of the lattice matched GaN$_x$Bi$_y$As$_{1-x-y}$/GaAs quaternary alloy quantum well (QW) based on the 16-band k$\cdot$p model. We have taken into consideration the interactions between the N and Bi impurity states with the host material based on the band anticrossing (BAC) and valence band anticrossing (VBAC) model. The optical gain calculation is based on the density matrix theory. We have considered different lattice matched GaNBiAs QW cases and studied their energy dispersion curves, optical gain spectrum, maximum optical gain and differential gain; and compared their performances based on these factors. The thickness and composition of these QWs were varied in order to keep the emission peak fixed at 1.55 $μ$m. The well thickness has an effect on the spectral width of the gain curves. On the other hand, a variation in the injection carrier density has different effects on the maximum gain and differential gain of QWs of varying thicknesses. Among the cases studied, we found that the 6.3 nm thick GaN$_3$Bi$_{5.17}$As$_{91.83}$ lattice matched QW was most suited for 1.55 $μ$m (0.8 eV) GaAs-based photonic applications. △ Less

Submitted 21 June, 2017; originally announced June 2017.

Comments: Accepted in AIP Journal of Applied Physics

Inhomogeneity parameter in designing an ion trap

Authors: Weikang Fan

Abstract: In designing an ion trap, geometry and rf source should be optimized such that the trap depth is maximized while the ion remain stable. In a quadrupole linear trap, stable parameters $a$ and $q$ are utilized frequently in describing the stability. However, in a surface trap, the trap have to be mapped to the linear quadrupole trap so that $a$ and $q$ can be evaluated. This work explains how to han… ▽ More In designing an ion trap, geometry and rf source should be optimized such that the trap depth is maximized while the ion remain stable. In a quadrupole linear trap, stable parameters $a$ and $q$ are utilized frequently in describing the stability. However, in a surface trap, the trap have to be mapped to the linear quadrupole trap so that $a$ and $q$ can be evaluated. This work explains how to handle them for surface trap designing and how the geometry and rf source affect it. We conclude that the $q$ parameter should be 0.2~0.22 so that the trap is stable. △ Less

Submitted 7 January, 2015; v1 submitted 26 December, 2014; originally announced December 2014.

Comments: This paper has been withdrawn by the author as the supervisor consider it not suitable to publish

Engineering large-scale entanglement in the quantum optical frequency comb: influence of the quasiphasematching bandwidth and of dispersion

Authors: Pei Wang, Wenjiang Fan, Olivier Pfister

Abstract: One of the most scalable methods for continuous-variable quantum computing is to encode quantum information in the quantum optical frequency comb generated by an optical parametric oscillator (OPO). The scalability is limited by the quasiphasematching bandwidth and dispersion of the OPO. We study both factors in this article. The results show that 3200 qumodes are probably entangled in our recent… ▽ More One of the most scalable methods for continuous-variable quantum computing is to encode quantum information in the quantum optical frequency comb generated by an optical parametric oscillator (OPO). The scalability is limited by the quasiphasematching bandwidth and dispersion of the OPO. We study both factors in this article. The results show that 3200 qumodes are probably entangled in our recent demonstration of a record 60-qumode entanglement. △ Less

Submitted 14 January, 2015; v1 submitted 26 March, 2014; originally announced March 2014.

Comments: Much expanded version from the previous one: now including the effects of dispersion. 7 pages, 6 figures, submitted for publication

Heavily Enhanced Dynamic Stark Shift in a System of Bose Einstein Condensation of Photons

Authors: Weikang Fan, Miao Yin, Ze Cheng

Abstract: The dynamic Stark shift of a high-lying atom in a system of Bose Einstein condensation (BEC) of photons is discussed within the framework of nonrelativistic quantum electrodynamics (QED) theory. It is found that the Stark shift of an atom in BEC of photons is modified by a temperature dependent factor, compared to that in a normal two-dimensional photonic fluid. In photonic BEC, the value of Stark… ▽ More The dynamic Stark shift of a high-lying atom in a system of Bose Einstein condensation (BEC) of photons is discussed within the framework of nonrelativistic quantum electrodynamics (QED) theory. It is found that the Stark shift of an atom in BEC of photons is modified by a temperature dependent factor, compared to that in a normal two-dimensional photonic fluid. In photonic BEC, the value of Stark shift is always greater than that in two-dimensional free space. Physical origin of the phenomenon is presented and potential application is also discussed. △ Less

Submitted 20 May, 2013; v1 submitted 29 January, 2013; originally announced January 2013.

Comments: 7 pages, 2figures

Long-range adiabatic quantum state transfer through a tight-binding chain as a quantum data bus

Authors: Bing Chen, Wei Fan, Yan Xu, Zhao-yang Chen, Xun-li Feng, C. H. Oh

Abstract: We introduce a scheme based on adiabatic passage that allows for long-range quantum communication through tight-binding chain with always-on interaction. By adiabatically varying the external gate voltage applied on the system, the electron can be transported from the sender's dot to the aim one.We numerically solve the Schrödinger equation for a system with a given number of quantum dots. It is s… ▽ More We introduce a scheme based on adiabatic passage that allows for long-range quantum communication through tight-binding chain with always-on interaction. By adiabatically varying the external gate voltage applied on the system, the electron can be transported from the sender's dot to the aim one.We numerically solve the Schrödinger equation for a system with a given number of quantum dots. It is shown that this scheme is a simple and efficient protocol to coherently manipulate the population transfer under suitable gate pulses. The dependence of the energy gap and the transfer time on system parameters is analyzed and shown numerically. Our method provides a guidance for future realization of adiabatic quantum state transfer in experiments. △ Less

Submitted 8 July, 2012; originally announced July 2012.

Comments: 7 pages, 5 figures. arXiv admin note: text overlap with arXiv:1206.6716

Journal ref: Physical Review A 86, 012302 (2012)

Long-range adiabatic quantum state transfer through a linear array of quantum dots

Authors: Bing Chen, Q. H. Shen, Wei Fan, Yan Xu

Abstract: We introduce an adiabatic long-range quantum communication proposal based on a quantum dot array. By adiabatically varying the external gate voltage applied on the system, the quantum information encoded in the electron can be transported from one end dot to another. We numerically solve the Schrödinger equation for a system with a given number of quantum dots. It is shown that this scheme is a si… ▽ More We introduce an adiabatic long-range quantum communication proposal based on a quantum dot array. By adiabatically varying the external gate voltage applied on the system, the quantum information encoded in the electron can be transported from one end dot to another. We numerically solve the Schrödinger equation for a system with a given number of quantum dots. It is shown that this scheme is a simple and efficient protocol to coherently manipulate the population transfer under suitable gate pulses. The dependence of the energy gap and the transfer time on system parameters is analyzed and shown numerically. We also investigate the adiabatic passage in a more realistic system in the presence of inevitable fabrication imperfections. This method provides guidance for future realizations of adiabatic quantum state transfer in experiments. △ Less

Submitted 28 June, 2012; originally announced June 2012.

Comments: 7 pages, 7 figures

Journal ref: Sci China-Phys Mech Astron, 2012, 55: 1-6

Quantum phase transition of two-mode Bose-Einstein condensates with an entanglement order parameter

Authors: Wei Fan, Yan Xu. Bing Chen, Zhaoyang Chen, Xunli Feng, C. H. Oh

Abstract: The ground state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum phase transition approach. The entanglement measure is taken as the order parameter and this is a non-local order parameter, which is different from the conventional order parameter of the Mott insulator-superfluid phase transitions. For this non-local order parameter, scaling behavior correspo… ▽ More The ground state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum phase transition approach. The entanglement measure is taken as the order parameter and this is a non-local order parameter, which is different from the conventional order parameter of the Mott insulator-superfluid phase transitions. For this non-local order parameter, scaling behavior corresponding to a continuous phase transition is obtained and a power-law divergence near the critical region follows it. This scaling behavior of quantum entanglement is analyzed by the finite-size scaling and the critical exponents are obtained as $ν= 1.01$ and $γ= 0.86$. A close connection between quantum fluctuations and the phase transition of entanglement is also obtained. △ Less

Submitted 19 January, 2012; v1 submitted 13 August, 2011; originally announced August 2011.

Comments: 4 figures, accepted by Phys. Rev. A, 2012

Journal ref: Phys. Rev. A 85, 013645 (2012)

Quantum corrections to the dynamics of the Bose-Einstein condensate in a double-well potential

Authors: Wei Fan, Yan Xu, Bing Chen

Abstract: The dynamics of the Bose-Einstein condensate (BEC) in a double-well potential is of- ten investigated under the mean-field theory (MFT). This works successfully for large particle numbers with dynamical stability. But for dynamical instabilities, quantum cor- rections to the MFT becomes important [Phys.Rev.A 64, 013605(2001)]. Recently the adiabatic dynamics of the double-well BEC is investigated… ▽ More The dynamics of the Bose-Einstein condensate (BEC) in a double-well potential is of- ten investigated under the mean-field theory (MFT). This works successfully for large particle numbers with dynamical stability. But for dynamical instabilities, quantum cor- rections to the MFT becomes important [Phys.Rev.A 64, 013605(2001)]. Recently the adiabatic dynamics of the double-well BEC is investigated under the MFT in terms of a dark variable [Phys.Rev.A 81, 043621(2010)], which generalizes the adiabatic passage techniques in quantum optics to the nonlinear matter-wave case. We give a fully quan- tized version of it using second-quantization and introduce new correction terms from higher order interactions beyond the on-site interaction, which are interactions between the tunneling particle and the particle in the well and interactions between the tunneling particles. If only the on-site interaction is considered, this reduces to the usual two-mode BEC. △ Less

Submitted 22 February, 2012; v1 submitted 20 March, 2011; originally announced March 2011.

Comments: accepted by Internal Journal of Modern Physics B recently

Solve the Master Equation by Python-An Introduction to the Python Computing Environment

Authors: Wei Fan, Yan Xu, Bing Chen, Qianqian Ye

Abstract: A brief introduction to the Python computing environment is given. By solving the master equation encountered in quantum transport, we give an example of how to solve the ODE problems in Python. The ODE solvers used are the ZVODE routine in Scipy and the bsimp solver in GSL. For the former, the equation can be in its complex-valued form, while for the latter, it has to be rewritten to a real-value… ▽ More A brief introduction to the Python computing environment is given. By solving the master equation encountered in quantum transport, we give an example of how to solve the ODE problems in Python. The ODE solvers used are the ZVODE routine in Scipy and the bsimp solver in GSL. For the former, the equation can be in its complex-valued form, while for the latter, it has to be rewritten to a real-valued form. The focus is on the detailed workflow of the implementation process, rather than on the syntax of the python language, with the hope to help readers simulate their own models in Python. △ Less

Submitted 10 May, 2011; v1 submitted 1 March, 2011; originally announced March 2011.

Comments: 26 pages, 5 figures

Adiabatic quantum state transfer in non-uniform triple-quantum-dot system

Authors: Bing Chen, Wei Fan, Yan Xu

Abstract: We introduce an adiabatic quantum state transfer scheme in a non-uniform coupled triple-quantum-dot system. By adiabatically varying the external gate voltage applied on the sender and receiver, the electron can be transferred between them with high fidelity. By numerically solving the master equation for a system with always-on interaction, it is indicated that the transfer fidelity depends on th… ▽ More We introduce an adiabatic quantum state transfer scheme in a non-uniform coupled triple-quantum-dot system. By adiabatically varying the external gate voltage applied on the sender and receiver, the electron can be transferred between them with high fidelity. By numerically solving the master equation for a system with always-on interaction, it is indicated that the transfer fidelity depends on the ration between the peak voltage and the maximum coupling constants. The effect of coupling mismatch on the transfer fidelity is also investigated and it is shown that there is a relatively large tolerance range to permit high fidelity quantum state transfer. △ Less

Submitted 20 October, 2010; v1 submitted 18 October, 2010; originally announced October 2010.

Comments: 6 pages, 5 figures

Journal ref: Phys. Rev. A 83, 014301 (2011)