We summarize recent theoretical results for the signatures of strongly correlated ultra-cold ferm... more We summarize recent theoretical results for the signatures of strongly correlated ultra-cold fermions in optical lattices. In particular, we focus on: collective mode calculations, where a sharp decrease in collective mode frequency is predicted at the onset of the Mott metal-insulator transition; and correlation functions at finite temperature, where we employ a new exact technique that applies the stochastic gauge technique with a Gaussian operator basis.
ABSTRACT Experiments with ultracold atoms present an outstanding opportunity for implementing nov... more ABSTRACT Experiments with ultracold atoms present an outstanding opportunity for implementing novel tests of theories of strongly correlated fermions. We present three new theoretical methods to treat these systems. Luttinger liquid theory and the exact one-dimensional solutions let us calculate collective mode frequencies at the metal-insulator transition in a lattice. A new Gaussian phase-space method for fermion systems can be used to simulate finite temperature atomic correlations. Finally, we introduce an approximate diagrammatic technique which correctly includes molecule-molecule interactions, and gives an accurate, quantitative theory of the BEC-BCS crossover regime.
We show that violation of genuine multipartite Bell inequalities can be obtained with sampled, pr... more We show that violation of genuine multipartite Bell inequalities can be obtained with sampled, probabilistic phase space methods. These genuine Bell violations cannot be replicated if any part of the system is described by a local hidden variable theory. The Bell violations are simulated probabilistically using quantum phase-space representations. We treat mesoscopically large Greenberger-Horne-Zeilinger (GHZ) states having up to 60 qubits, using both a multipartite SU(2) Q-representation and the positive P-representation. Surprisingly, we find that sampling with phase-space distributions can be exponentially faster than experiment. This is due to the classical parallelism inherent in the simulation of quantum measurements using phase-space methods. Our probabilistic sampling method predicts a contradiction with local realism of "Schrödinger-cat" states that can be realized as a GHZ spin state, either in ion traps or with photonic qubits. We also present a quantum simulation of the observed super-decoherence of the ion-trap "cat" state, using a phenomenological noise model.
Quantum simulations of Bell inequality violations are numerically obtained using probabilistic ph... more Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase space methods. Here, the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. A number of states violating Bell inequalities are simulated, demonstrating that these quantum paradoxes can be treated with probabilistic methods. Further, we carry out a simulation of quantum dynamics, by simulating the evolution of the Bell state formed via parametric down-conversion, and discuss multi-mode generalizations.
We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the EPR-ste... more We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the EPR-steering paradox between groups of atoms in two separated potential wells in a Bose-Einstein condensate (BEC). We present experimental criteria for this form of entanglement, and propose experimental strategies using adiabatic cooling to the ground state. These approaches use either two or four spatial and/or internal modes. We also present higher order criteria that act as signatures to detect the multiparticle entanglement present in this system. We point out the difference between spatial entanglement using separated detectors, and other types of entanglement that do not require spatial separation. The four-mode approach with two spatial and two internal modes results in an entanglement signature with spatially separated detectors, conceptually similar to the original EPR paradox.
We review recent developments in the theory of quantum dynamics in ultracold atomic physics, incl... more We review recent developments in the theory of quantum dynamics in ultracold atomic physics, including exact techniques and methods based on phase-space mappings that are applicable when the complexity becomes exponentially large. Phase-space representations include the truncated Wigner, positive-P and general Gaussian operator representations which can treat both bosons and fermions. These phase-space methods include both traditional approaches using a phase-space of classical dimension, and more recent methods that use a non-classical phase-space of increased dimensionality. Examples used include quantum Einstein-Podolsky-Rosen (EPR) entanglement of a four-mode BEC, time-reversal tests of dephasing in single-mode traps, BEC quantum collisions with up to 106 modes and 105 interacting particles, quantum interferometry in a multi-mode trap with nonlinear absorption, and the theory of quantum entropy in phase-space. We also treat the approach of variational optimization of the samplin...
We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein co... more We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein condensate atom interferometer with nonlinear losses. We use stochastic equations in a truncated Wigner representation to treat quantum noise. Our approach includes the multi-mode spatial evolution of spinor components and describes the many-body dynamics of a mesoscopic quantum system.
Quantum simulations of Bell inequality violations are numerically obtained using probabilistic ph... more Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase space methods. Here, the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. A number of states violating Bell inequalities are simulated, demonstrating that these quantum paradoxes can be treated with probabilistic methods. Further, we carry out a simulation of quantum dynamics, by simulating the evolution of the Bell state formed via parametric down-conversion, and discuss multi-mode generalizations.
We propose to generate Einstein-Podolsky-Rosen (EPR) entanglement between groups of atoms in a tw... more We propose to generate Einstein-Podolsky-Rosen (EPR) entanglement between groups of atoms in a two-well Bose-Einstein condensate using a dynamical process similar to that employed in quantum optics. A local nonlinear S-wave scattering interaction has the effect of creating spin squeezing at each well, while a tunneling coupling, analogous to a beam splitter in optics, introduces an interference between these fields that causes interwell entanglement. We consider two internal modes at each well so that the entanglement can be detected by measuring a reduction in the variances of the sums of local Schwinger spin observables. As is typical of continuous variable (CV) entanglement, the entanglement is predicted to increase with atom number. It becomes sufficiently strong at higher numbers of atoms so that the EPR paradox and steering nonlocality can be realized. The entanglement is predicted using an analytical approach and, for larger atom numbers, using stochastic simulations based on...
We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside... more We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside a cavity resonant for all modes but the initial pump. Due to the resonant cascade design, the OPO presents twoχ (2) -level oscillation thresholds that are therefore much lower than for aχ (3) OPO. This is promising for reaching the regime of an effective third-order nonlinearity well above both thresholds. Such aχ (2) cascaded device also has potential applications in frequency conversion to far-infrared regimes. But, most importantly, it can generate novel multipartite quantum correlations in the output radiation, which represent a step beyond squeezed or entangled light. The output can be highly non-Gaussian and therefore not describable by any semiclassical model. In this paper, we derive quantum stochastic equations in the positive-Prepresentation and undertake an analysis of steady-state and dynamical properties of this system.
We investigate polarization squeezing of ultrashort pulses in optical fiber, over a wide range of... more We investigate polarization squeezing of ultrashort pulses in optical fiber, over a wide range of input energies and fiber lengths. Comparisons are made between experimental data and quantum dynamical simulations to find good quantitative agreement. The numerical calculations, performed using both truncated Wigner and exact +P phase-space methods, include nonlinear and stochastic Raman effects, through coupling to phonon variables. The simulations reveal that excess phase noise, such as from depolarizing guided acoustic wave Brillouin scattering, affects squeezing at low input energies, while Raman effects cause a marked deterioration of squeezing at higher energies and longer fiber lengths. We also calculate the optimum fiber length for maximum squeezing.
We introduce an approximate phase-space technique to simulate the quantum dynamics of interacting... more We introduce an approximate phase-space technique to simulate the quantum dynamics of interacting bosons. With the future goal of treating Bose-Einstein condensate systems, the method is designed for systems with a natural separation into highly occupied (condensed) modes and lightly occupied modes. The method self-consistently uses the Wigner representation to treat highly occupied modes and the positive-P representation for lightly occupied modes. In this method, truncation of higher-derivative terms from the Fokker-Planck equation is usually necessary. However, at least in the cases investigated here, the resulting systematic error, over a finite time, vanishes in the limit of large Wigner occupation numbers. We tested the method on a system of two interacting anharmonic oscillators, with high and low occupations, respectively. The Hybrid method successfully predicted atomic quadratures to a useful simulation time 60 times longer than that of the positive-P method. The truncated ...
Laser Spectroscopy - Proceedings of the XVI International Conference, 2004
We review progress towards direct simulation of quantum dynamics in many-body systems, using rece... more We review progress towards direct simulation of quantum dynamics in many-body systems, using recently developed stochastic gauge techniques. We consider master equations, canonical ensemble calculations and reversible quantum dynamics are compared, as well the general question of strategies for choosing the gauge.
We summarize recent theoretical results for the signatures of strongly correlated ultra-cold ferm... more We summarize recent theoretical results for the signatures of strongly correlated ultra-cold fermions in optical lattices. In particular, we focus on: collective mode calculations, where a sharp decrease in collective mode frequency is predicted at the onset of the Mott metal-insulator transition; and correlation functions at finite temperature, where we employ a new exact technique that applies the stochastic gauge technique with a Gaussian operator basis.
ABSTRACT Experiments with ultracold atoms present an outstanding opportunity for implementing nov... more ABSTRACT Experiments with ultracold atoms present an outstanding opportunity for implementing novel tests of theories of strongly correlated fermions. We present three new theoretical methods to treat these systems. Luttinger liquid theory and the exact one-dimensional solutions let us calculate collective mode frequencies at the metal-insulator transition in a lattice. A new Gaussian phase-space method for fermion systems can be used to simulate finite temperature atomic correlations. Finally, we introduce an approximate diagrammatic technique which correctly includes molecule-molecule interactions, and gives an accurate, quantitative theory of the BEC-BCS crossover regime.
We show that violation of genuine multipartite Bell inequalities can be obtained with sampled, pr... more We show that violation of genuine multipartite Bell inequalities can be obtained with sampled, probabilistic phase space methods. These genuine Bell violations cannot be replicated if any part of the system is described by a local hidden variable theory. The Bell violations are simulated probabilistically using quantum phase-space representations. We treat mesoscopically large Greenberger-Horne-Zeilinger (GHZ) states having up to 60 qubits, using both a multipartite SU(2) Q-representation and the positive P-representation. Surprisingly, we find that sampling with phase-space distributions can be exponentially faster than experiment. This is due to the classical parallelism inherent in the simulation of quantum measurements using phase-space methods. Our probabilistic sampling method predicts a contradiction with local realism of "Schrödinger-cat" states that can be realized as a GHZ spin state, either in ion traps or with photonic qubits. We also present a quantum simulation of the observed super-decoherence of the ion-trap "cat" state, using a phenomenological noise model.
Quantum simulations of Bell inequality violations are numerically obtained using probabilistic ph... more Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase space methods. Here, the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. A number of states violating Bell inequalities are simulated, demonstrating that these quantum paradoxes can be treated with probabilistic methods. Further, we carry out a simulation of quantum dynamics, by simulating the evolution of the Bell state formed via parametric down-conversion, and discuss multi-mode generalizations.
We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the EPR-ste... more We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the EPR-steering paradox between groups of atoms in two separated potential wells in a Bose-Einstein condensate (BEC). We present experimental criteria for this form of entanglement, and propose experimental strategies using adiabatic cooling to the ground state. These approaches use either two or four spatial and/or internal modes. We also present higher order criteria that act as signatures to detect the multiparticle entanglement present in this system. We point out the difference between spatial entanglement using separated detectors, and other types of entanglement that do not require spatial separation. The four-mode approach with two spatial and two internal modes results in an entanglement signature with spatially separated detectors, conceptually similar to the original EPR paradox.
We review recent developments in the theory of quantum dynamics in ultracold atomic physics, incl... more We review recent developments in the theory of quantum dynamics in ultracold atomic physics, including exact techniques and methods based on phase-space mappings that are applicable when the complexity becomes exponentially large. Phase-space representations include the truncated Wigner, positive-P and general Gaussian operator representations which can treat both bosons and fermions. These phase-space methods include both traditional approaches using a phase-space of classical dimension, and more recent methods that use a non-classical phase-space of increased dimensionality. Examples used include quantum Einstein-Podolsky-Rosen (EPR) entanglement of a four-mode BEC, time-reversal tests of dephasing in single-mode traps, BEC quantum collisions with up to 106 modes and 105 interacting particles, quantum interferometry in a multi-mode trap with nonlinear absorption, and the theory of quantum entropy in phase-space. We also treat the approach of variational optimization of the samplin...
We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein co... more We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein condensate atom interferometer with nonlinear losses. We use stochastic equations in a truncated Wigner representation to treat quantum noise. Our approach includes the multi-mode spatial evolution of spinor components and describes the many-body dynamics of a mesoscopic quantum system.
Quantum simulations of Bell inequality violations are numerically obtained using probabilistic ph... more Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase space methods. Here, the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. A number of states violating Bell inequalities are simulated, demonstrating that these quantum paradoxes can be treated with probabilistic methods. Further, we carry out a simulation of quantum dynamics, by simulating the evolution of the Bell state formed via parametric down-conversion, and discuss multi-mode generalizations.
We propose to generate Einstein-Podolsky-Rosen (EPR) entanglement between groups of atoms in a tw... more We propose to generate Einstein-Podolsky-Rosen (EPR) entanglement between groups of atoms in a two-well Bose-Einstein condensate using a dynamical process similar to that employed in quantum optics. A local nonlinear S-wave scattering interaction has the effect of creating spin squeezing at each well, while a tunneling coupling, analogous to a beam splitter in optics, introduces an interference between these fields that causes interwell entanglement. We consider two internal modes at each well so that the entanglement can be detected by measuring a reduction in the variances of the sums of local Schwinger spin observables. As is typical of continuous variable (CV) entanglement, the entanglement is predicted to increase with atom number. It becomes sufficiently strong at higher numbers of atoms so that the EPR paradox and steering nonlocality can be realized. The entanglement is predicted using an analytical approach and, for larger atom numbers, using stochastic simulations based on...
We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside... more We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside a cavity resonant for all modes but the initial pump. Due to the resonant cascade design, the OPO presents twoχ (2) -level oscillation thresholds that are therefore much lower than for aχ (3) OPO. This is promising for reaching the regime of an effective third-order nonlinearity well above both thresholds. Such aχ (2) cascaded device also has potential applications in frequency conversion to far-infrared regimes. But, most importantly, it can generate novel multipartite quantum correlations in the output radiation, which represent a step beyond squeezed or entangled light. The output can be highly non-Gaussian and therefore not describable by any semiclassical model. In this paper, we derive quantum stochastic equations in the positive-Prepresentation and undertake an analysis of steady-state and dynamical properties of this system.
We investigate polarization squeezing of ultrashort pulses in optical fiber, over a wide range of... more We investigate polarization squeezing of ultrashort pulses in optical fiber, over a wide range of input energies and fiber lengths. Comparisons are made between experimental data and quantum dynamical simulations to find good quantitative agreement. The numerical calculations, performed using both truncated Wigner and exact +P phase-space methods, include nonlinear and stochastic Raman effects, through coupling to phonon variables. The simulations reveal that excess phase noise, such as from depolarizing guided acoustic wave Brillouin scattering, affects squeezing at low input energies, while Raman effects cause a marked deterioration of squeezing at higher energies and longer fiber lengths. We also calculate the optimum fiber length for maximum squeezing.
We introduce an approximate phase-space technique to simulate the quantum dynamics of interacting... more We introduce an approximate phase-space technique to simulate the quantum dynamics of interacting bosons. With the future goal of treating Bose-Einstein condensate systems, the method is designed for systems with a natural separation into highly occupied (condensed) modes and lightly occupied modes. The method self-consistently uses the Wigner representation to treat highly occupied modes and the positive-P representation for lightly occupied modes. In this method, truncation of higher-derivative terms from the Fokker-Planck equation is usually necessary. However, at least in the cases investigated here, the resulting systematic error, over a finite time, vanishes in the limit of large Wigner occupation numbers. We tested the method on a system of two interacting anharmonic oscillators, with high and low occupations, respectively. The Hybrid method successfully predicted atomic quadratures to a useful simulation time 60 times longer than that of the positive-P method. The truncated ...
Laser Spectroscopy - Proceedings of the XVI International Conference, 2004
We review progress towards direct simulation of quantum dynamics in many-body systems, using rece... more We review progress towards direct simulation of quantum dynamics in many-body systems, using recently developed stochastic gauge techniques. We consider master equations, canonical ensemble calculations and reversible quantum dynamics are compared, as well the general question of strategies for choosing the gauge.
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Papers by Peter Drummond