Showing papers in "Physical Review A in 2006"
TL;DR: It is shown how to efficiently simulate a quantum many-body system with tree structure when its entanglement (Schmidt number) is small for any bipartite split along an edge of the tree.
Abstract: We show how to efficiently simulate a quantum many-body system with tree structure when its entanglement (Schmidt number) is small for any bipartite split along an edge of the tree. As an application, we show that any one-way quantum computation on a tree graph can be efficiently simulated with a classical computer.
500 citations
TL;DR: In this article, the authors present a practical way to reduce the maximal information gain that an adversary can gain using Trojan-horse attacks on quantum key distribution systems, provided that enough additional privacy amplification is applied to the data.
Abstract: General Trojan-horse attacks on quantum-key-distribution systems, i.e., attacks on Alice or Bob's system via the quantum channel, are analyzed. We illustrate the power of such attacks with today's technology and conclude that all systems must implement active counter measures. In particular, all systems must include an auxiliary detector that monitors any incoming light. We show that such counter measures can be efficient, provided that enough additional privacy amplification is applied to the data. We present a practical way to reduce the maximal information gain that an adversary can gain using Trojan-horse attacks. This does reduce the security analysis of the two-way plug-and-play implementation to those of the standard one-way systems.
471 citations
TL;DR: In three dimensions, this work provides evidence, in the form a simple mean field theory, that the Hamiltonian gives rise to a system which is self-correcting, robust to noise without external intervening quantum error-correction procedures.
Abstract: The most general method for encoding quantum information is not to encode the information into a subspace of a Hilbert space, but to encode information into a subsystem of a Hilbert space. Recently this notion has led to a more general notion of quantum error correction known as operator quantum error correction. In standard quantum error-correcting codes, one requires the ability to apply a procedure which exactly reverses on the error-correcting subspace any correctable error. In contrast, for operator error-correcting subsystems, the correction procedure need not undo the error which has occurred, but instead one must perform corrections only modulo the subsystem structure. This does not lead to codes which differ from subspace codes, but does lead to recovery routines which explicitly make use of the subsystem structure. Here we present two examples of such operator error-correcting subsystems. These examples are motivated by simple spatially local Hamiltonians on square and cubic lattices. In three dimensions we provide evidence, in the form a simple mean field theory, that our Hamiltonian gives rise to a system which is self-correcting. Such a system will be a natural high-temperature quantum memory, robust to noise without external intervening quantum error-correction procedures.
455 citations
TL;DR: In this article, it is shown that there is a class of $W$ states that can be used for perfect teleportation and super-dense coding, and that these states can be classified into two categories: GHZ states and W$ states.
Abstract: True tripartite entanglement of the state of a system of three qubits can be classified on the basis of stochastic local operations and classical communications. Such states can be classified into two categories: GHZ states and $W$ states. It is known that GHZ states can be used for teleportation and superdense coding, but the prototype $W$ state cannot be. However, we show that there is a class of $W$ states that can be used for perfect teleportation and superdense coding.
432 citations
TL;DR: The security of the secure direct communication protocol based on the secret transmitting order of particles recently proposed is found to be insecure if an eavesdropper, say Eve, wants to steal the secret message with Trojan horse attack strategies.
Abstract: We analyzed the security of the secure direct communication protocol based on the secret transmitting order of particles recently proposed by Zhu, Xia, Fan, and Zhang[Phys. Rev. A 73, 022338 (2006)] and found that this scheme is insecure if an eavesdropper, say Eve, wants to steal the secret message with Trojan horse attack strategies. The vital loophole in this scheme is that the two authorized users check the security of their quantum channel only once. Eve can insert another spy photon, an invisible photon, or a delay one in each photon which the sender Alice sends to the receiver Bob, and capture the spy photon when it returns from Bob to Alice. After the authorized users check the security, Eve can obtain the secret message according to the information about the transmitting order published by Bob. Finally, we present a possible improvement of this protocol.
430 citations
TL;DR: In this paper, the authors analyzed the entanglement between two modes of a free Dirac field as seen by two relatively accelerated parties and showed that the state always remains entangled to a degree and can be used in quantum information tasks, such as teleportation.
Abstract: We analyze the entanglement between two modes of a free Dirac field as seen by two relatively accelerated parties. The entanglement is degraded by the Unruh effect and asymptotically reaches a nonvanishing minimum value in the infinite acceleration limit. This means that the state always remains entangled to a degree and can be used in quantum information tasks, such as teleportation, between parties in relative uniform acceleration. We analyze our results from the point of view afforded by the phenomenon of entanglement sharing and in terms of recent results in the area of multiqubit complementarity.
403 citations
TL;DR: In this paper, a simple, robust, and ultrabright parametric down-conversion source of polarization-entangled photons based on a polarization Sagnac interferometer was demonstrated.
Abstract: We demonstrate a simple, robust, and ultrabright parametric down-conversion source of polarization-entangled photons based on a polarization Sagnac interferometer. Bidirectional pumping in type-II phase-matched periodically poled $\mathrm{K}\mathrm{Ti}\mathrm{O}\mathrm{P}{\mathrm{O}}_{4}$ yields a measured flux of 5000 polarization-entangled pairs/s/mW of pump power in a $1\text{\ensuremath{-}}\mathrm{nm}$ bandwidth at 96.8% quantum-interference visibility. The common-path arrangement of the Sagnac interferometer eliminates the need for phase stabilization for the biphoton output state.
363 citations
TL;DR: In this paper, the authors theoretically study measurement-induced dephasing of a superconducting qubit in the circuit QED architecture and compare the results to those obtained experimentally by Schuster et al.
Abstract: We theoretically study measurement-induced dephasing of a superconducting qubit in the circuit QED architecture and compare the results to those obtained experimentally by Schuster et al. [Phys. Rev. Lett. 94, 123602 (2005)]. Strong coupling of the qubit to the resonator leads to a significant ac Stark shift of the qubit transition frequency. As a result, quantum fluctuations in the photon number populating the resonator cause dephasing of the qubit. We find good agreement between the predicted line shape of the qubit spectrum and the experimental results. Furthermore, in the strongly dispersive limit, where the Stark shift per photon is large compared to the cavity decay rate and the qubit linewidth, we predict that the qubit spectrum will be split into multiple peaks, with each peak corresponding to a different number of photons in the cavity.
363 citations
TL;DR: For the Bennett-Brassard 1984 (BB84) protocol, it is shown that if the efficiency mismatch between 0 and 1 detectors for some value of the control parameter gets large enough, Eve can construct a successful faked-states attack causing a quantum bit error rate lower than 11%.
Abstract: We suggest a type of attack on quantum cryptosystems that exploits variations in detector efficiency as a function of a control parameter accessible to an eavesdropper. With gated single-photon detectors, this control parameter can be the timing of the incoming pulse. When the eavesdropper sends short pulses using the appropriate timing so that the two gated detectors in Bob's setup have different efficiencies, the security of quantum key distribution can be compromised. Specifically, we show for the Bennett-Brassard 1984 (BB84) protocol that if the efficiency mismatch between 0 and 1 detectors for some value of the control parameter gets large enough (roughly 15:1 or larger), Eve can construct a successful faked-states attack causing a quantum bit error rate lower than 11%. We also derive a general security bound as a function of the detector sensitivity mismatch for the BB84 protocol. Experimental data for two different detectors are presented, and protection measures against this attack are discussed.
351 citations
TL;DR: In this article, the authors studied transient entanglement between two qubits coupled collectively to a multimode vacuum field, assuming that the two-qubit system is initially prepared in an entangled state produced by two-photon coherences.
Abstract: In a recent paper Yu and Eberly [Phys. Rev. Lett. 93, 140404 (2004)] have shown that two initially entangled and afterward not interacting qubits can become completely disentangled in a finite time. We study transient entanglement between two qubits coupled collectively to a multimode vacuum field, assuming that the two-qubit system is initially prepared in an entangled state produced by the two-photon coherences, and find the unusual feature that the irreversible spontaneous decay can lead to a revival of the entanglement that has already been destroyed. The results show that this feature is independent of the coherent dipole-dipole interaction between the atoms but it depends critically on whether or not collective damping is present.
339 citations
TL;DR: In this article, it was shown that for any distribution, the properties of non-local, no arbitrarily shareable, and positive secrecy content are equivalent, and that all these properties are solely a consequence of the no-signaling principle and non-locality.
Abstract: This article identifies a series of properties common to all theories that do not allow for superluminal signaling and predict the violation of Bell inequalities. Intrinsic randomness, uncertainty due to the incompatibility of two observables, monogamy of correlations, impossibility of perfect cloning, privacy of correlations, bounds in the shareability of some states; all these phenomena are solely a consequence of the no-signaling principle and nonlocality. In particular, it is shown that for any distribution, the properties of (i) nonlocal, (ii) no arbitrarily shareable, and (iii) positive secrecy content are equivalent.
TL;DR: In this paper, a theoretical interpretation of a recent experiment presented by Zwierlein et al. on the density profile of Fermi gases with unbalanced spin populations is presented.
Abstract: We present a theoretical interpretation of a recent experiment presented by Zwierlein et al. [Nature (London) 442, 54 (2006)] on the density profile of Fermi gases with unbalanced spin populations. We show that in the regime of strong interaction, the boundaries of the three phases observed by Zwierlein et al. can be characterized by two dimensionless numbers ${\ensuremath{\eta}}_{\ensuremath{\alpha}}$ and ${\ensuremath{\eta}}_{\ensuremath{\beta}}$. Using a combination of a variational treatment and a study of the experimental results, we infer rather precise bounds for these two parameters.
TL;DR: In this paper, it was shown that the causal, non-instantaneous behavior of any single-photon sensitive nonlinearity is enough to preclude such a high-fidelity operation.
Abstract: By embedding an atom capable of electromagnetically induced transparency inside an appropriate photonic-crystal microcavity it may become possible to realize an optical nonlinearity that can impart a $\ensuremath{\pi}$-rad-peak phase shift in response to a single-photon excitation. Such a device, if it operated at high fidelity, would then complete a universal gate set for all-optical quantum computation. It is shown here that the causal, noninstantaneous behavior of any ${\ensuremath{\chi}}^{(3)}$ nonlinearity is enough to preclude such a high-fidelity operation. In particular, when a single-photon-sensitive ${\ensuremath{\chi}}^{(3)}$ nonlinearity has a response time that is much shorter than the duration of the quantum computer's single-photon pulses, essentially no overall phase shift is imparted to these pulses by cross-phase modulation. Conversely, when this nonlinearity has a response time that is much longer than this pulse duration a single-photon pulse can induce a $\ensuremath{\pi}$-rad overall phase shift through cross-phase modulation, but the phase noise injected by the causal, noninstantaneous response function precludes this from being a high-fidelity operation.
TL;DR: In these protocols, the secret transmitting order of particles ensures the security of communication, and no secret messages are leaked even if the communication is interrupted for security.
Abstract: We propose the schemes of quantum secure direct communication based on a secret transmitting order of particles. In these protocols, the secret transmitting order of particles ensures the security of communication, and no secret messages are leaked even if the communication is interrupted for security. This strategy of security for communication is also generalized to a quantum dialogue. It not only ensures the unconditional security but also improves the efficiency of communication.
TL;DR: Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Renyi entropies, and analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in N-level systems.
Abstract: Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the R\'enyi entropies. The proof of these inequalities requires the use of the exact expression for the $(p,q)$-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in $N$-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.
TL;DR: In this article, the authors considered wave dynamics with small dispersion and showed that this provides a mechanism for the generation of a dispersive shock wave (DSW) in a Bose-Einstein condensate.
Abstract: A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.
TL;DR: In this paper, the authors describe the design and implementation of a two-dimensional optical lattice of double wells suitable for isolating and manipulating an array of individual pairs of atoms in a 2D lattice, where each pair of atoms can be placed in a double well with any of their four nearest neighbors.
Abstract: We describe the design and implementation of a two-dimensional optical lattice of double wells suitable for isolating and manipulating an array of individual pairs of atoms in an optical lattice. Atoms in the square lattice can be placed in a double well with any of their four nearest neighbors. The properties of the double well (the barrier height and relative energy offset of the paired sites) can be dynamically controlled. The topology of the lattice is phase stable against phase noise imparted by vibrational noise on mirrors. We demonstrate the dynamic control of the lattice by showing the coherent splitting of atoms from single wells into double wells and observing the resulting double-slit atom diffraction pattern. This lattice can be used to test controlled neutral atom motion among lattice sites and should allow for testing controlled two-qubit gates.
TL;DR: This is the first method to construct a product basis exhibiting nonlocality without entanglement with n parties each holding a system of dimension at least n-1 via a quantum circuit made of controlled discrete Fourier transform gates acting on the computational basis.
Abstract: We present a generic method to construct a product basis exhibiting nonlocality without entanglement with $n$ parties each holding a system of dimension at least $n\ensuremath{-}1$. This basis is generated via a quantum circuit made of controlled discrete Fourier transform gates acting on the computational basis. The simplicity of our quantum circuit allows for an intuitive understanding of this new type of nonlocality. We also show how this circuit can be used to construct unextendible product bases and their associated bound entangled states. To our knowledge, this is the first method which, given a general Hilbert space $\mathcal{H}={\ensuremath{\bigotimes}}_{i=1}^{n}{\mathcal{H}}_{{d}_{i}}$ with ${d}_{i}\ensuremath{\leqslant}n\ensuremath{-}1$, makes it possible to construct (i) a basis exhibiting nonlocality without entanglement, (ii) an unextendible product basis, and (iii) a bound entangled state.
TL;DR: In this paper, the theory of bosonic systems with multiple condensates is presented, providing a unified description of various model systems that are found in the literature, and the authors discuss how degeneracies, interactions, and symmetries conspire to give rise to this unusual behavior.
Abstract: We present the theory of bosonic systems with multiple condensates, providing a unified description of various model systems that are found in the literature. We discuss how degeneracies, interactions, and symmetries conspire to give rise to this unusual behavior. We show that as degeneracies multiply, so do the varieties of fragmentation, eventually leading to strongly correlated states with no trace of condensation.
TL;DR: In this article, a full quantum theory for the dissipative dynamics of an optical cavity in the ultrastrong light-matter coupling regime is presented, in which the vacuum Rabi frequency is a significant fraction of the active electronic transition frequency and the antiresonant terms of the light matter coupling play an important role.
Abstract: We present a full quantum theory for the dissipative dynamics of an optical cavity in the ultrastrong light-matter coupling regime, in which the vacuum Rabi frequency is a significant fraction of the active electronic transition frequency and the antiresonant terms of the light-matter coupling play an important role. In particular, our model can be applied to the case of intersubband transitions in doped semiconductor quantum wells embedded in a microcavity. The coupling of the intracavity photonic mode and of the electronic polarization to the external, frequency-dependent, dissipation baths is taken into account by means of quantum Langevin equations in the input-output formalism. In the case of a time-independent vacuum Rabi frequency, exact analytical expressions for the operators are obtained, which allows us to characterize the quantum dissipative response of the cavity to an arbitrary initial condition (vacuum, coherent field, thermal excitation). For a vacuum input in both the photonic and electronic polarization modes, the ground state of the cavity system is a two-mode squeezed vacuum state with a finite population in both photonic and electronic modes. These excitations are, however, virtual and cannot escape from the cavity: for a vacuum input, a vacuum output is found, without any trace of the intracavity squeezing. For a coherent photonic input the linear optical response spectra (reflectivity, absorption, transmission) have been studied, and signatures of the ultrastrong coupling have been identified in the asymmetric and peculiar anticrossing of the polaritonic eigenmodes. Finally, we have calculated the electroluminescence spectra in the case of an incoherent electronic input: the emission intensity in the ultrastrong coupling regime results in being significantly enhanced as compared to the case of an isolated quantum well without a surrounding cavity.
TL;DR: In this paper, the authors show how ultracold polar molecules, suggested as a new platform for quantum computation, can be manipulated to switch ''on'' and ''off'' their strong dipole-dipole interactions.
Abstract: We show how ultracold polar molecules, suggested as a new platform for quantum computation, can be manipulated to switch ``on'' and ``off'' their strong dipole-dipole interactions. This can be accomplished through selective excitation of states with considerably different dipole moments. We discuss different schemes for quantum gates using real molecules: CO, LiH, and CaF, as examples of polar molecules which are being experimentally studied at ultracold temperatures. These schemes can be realized in several recently proposed architectures.
TL;DR: In this article, it was shown that the entropy of entanglement and the distillability of regions with respect to the rest of a general harmonic lattice system in the ground or a thermal state scale at most as the boundary area of the region.
Abstract: We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions—corresponding to the Klein-Gordon case— upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 2005. These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block-banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.
TL;DR: In this paper, the authors analyzed hybrid absorptive-dispersive optical bistability and multistability behavior in a generic $N$-type atomic system driven by a degenerate probe field and a coherent coupling field by means of a unidirectional ring cavity.
Abstract: We analyze hybrid absorptive-dispersive optical bistability (OB) and multistability (OM) behavior in a generic $N$-type atomic system driven by a degenerate probe field and a coherent coupling field by means of a unidirectional ring cavity. We show that the OB can be controlled by adjusting the intensity and the detuning of the coupling field, and the OM can also be observed under the appropriate detuning. The influence of the atomic cooperation parameter on atomic OB behavior is also discussed.
TL;DR: In this paper, the authors highlight the role of Lie algebras and noncommutativity in the design of a compensating pulse sequence and investigate three common dispersions in NMR spectroscopy, namely the Larmor dispersion, rf inhomogeneity and strength of couplings between the spins.
Abstract: Finding control fields (pulse sequences) that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating dispersion in system dynamics is widely known and applied. In this paper, we make explicit the key aspects of the dynamics that makes such a compensation possible. We highlight the role of Lie algebras and noncommutativity in the design of a compensating pulse sequence. Finally, we investigate three common dispersions in NMR spectroscopy, namely the Larmor dispersion, rf inhomogeneity, and strength of couplings between the spins.
TL;DR: In this article, a method for efficient storage and recall of arbitrary nonstationary light fields, such as single photon time-bin qubits or intense fields, in optically dense atomic ensembles is proposed.
Abstract: We propose a method for efficient storage and recall of arbitrary nonstationary light fields, such as, for instance, single photon time-bin qubits or intense fields, in optically dense atomic ensembles. Our approach to quantum memory is based on controlled, reversible, inhomogeneous broadening and relies on a hidden time-reversal symmetry of the optical Bloch equations describing the propagation of the light field. We briefly discuss experimental realizations of our proposal.
TL;DR: In this article, the authors considered topological quantum computation with a particular class of anyons that are believed to exist in the fractional quantum Hall effect state at Landau-level filling fraction v = 5/2.
Abstract: We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the fractional quantum Hall effect state at Landau-level filling fraction v =5/2. Since the braid group representation describing the statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy nontopological operations such as direct short-range interactions between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for nontopological operations is above 14%. The total number of nontopological computational elements that one needs to simulate a quantum circuit with L gates scales as L(ln L)to the 3rd.
TL;DR: In this article, the dominant electric-dipole contributions are computed using accurate relativistic many-body techniques of atomic structure, and the resulting uncertainties in the $E1$ BBR shifts are large and substantially affect the projected $10}^{\ensuremath{-}18}$ fractional accuracy of the optical-lattice-based clocks.
Abstract: Blackbody radiation (BBR) shifts of the $^{3}P_{0}\text{\ensuremath{-}}^{1}S_{0}$ clock transition in the divalent atoms Mg, Ca, Sr, and Yb are evaluated. The dominant electric-dipole contributions are computed using accurate relativistic many-body techniques of atomic structure. At room temperatures, the resulting uncertainties in the $E1$ BBR shifts are large and substantially affect the projected ${10}^{\ensuremath{-}18}$ fractional accuracy of the optical-lattice-based clocks. A peculiarity of these clocks is that the characteristic BBR wavelength is comparable to the $^{3}P$ fine-structure intervals. To evaluate relevant $M1$ and $E2$ contributions, a theory of multipolar BBR shifts is developed. The resulting corrections, although presently masked by the uncertainties in the $E1$ contribution, are required at the ${10}^{\ensuremath{-}18}$ accuracy goal.
TL;DR: In this paper, an erbium-doped microlaser was demonstrated utilizing passive microdisk resonators on a silicon chip, which exhibited whisperinggallery-type modes with intrinsic optical quality factors of up to $6.
Abstract: An erbium-doped microlaser is demonstrated utilizing $\mathrm{Si}{\mathrm{O}}_{2}$ microdisk resonators on a silicon chip. Passive microdisk resonators exhibit whispering-gallery-type modes (WGM's) with intrinsic optical quality factors of up to $6\ifmmode\times\else\texttimes\fi{}{10}^{7}$ and were doped with trivalent erbium ions (peak concentration $\ensuremath{\sim}3.8\ifmmode\times\else\texttimes\fi{}{10}^{20}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$) using MeV ion implantation. Coupling to the fundamental WGM of the microdisk resonator was achieved by using a tapered optical fiber. Upon pumping of the $^{4}I_{15∕2}\ensuremath{\rightarrow}^{4}I_{13∕2}$ erbium transition at $1450\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, a gradual transition from spontaneous to stimulated emission was observed in the $1550\text{\ensuremath{-}}\mathrm{nm}$ band. Analysis of the pump-output power relation yielded a pump threshold of $43\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{W}$ and allowed measuring the spontaneous emission coupling factor: $\ensuremath{\beta}\ensuremath{\approx}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$.
TL;DR: In this paper, an improved two-mode model for double-well potentials was proposed, which explicitly includes a nonlinear interaction term, which has been given previously in the literature but not widely appreciated.
Abstract: In this work, our primary goal has been to explore the range of validity of two-mode models for Bose-Einstein condensates in double-well potentials. Our derivation, like others, uses symmetric and antisymmetric condensate basis functions for the Gross-Pitaevskii equation. In what we call an ``improved two-mode model'' (I2M), the tunneling coupling energy explicitly includes a nonlinear interaction term, which has been given previously in the literature but not widely appreciated. We show that when the atom number (and hence the extent of the wave function) in each well vary appreciably with time, the nonlinear interaction term produces a temporal change in the tunneling energy or rate, which has not previously been considered to our knowledge. In addition, we obtain a parameter, labeled ``interaction tunneling,'' that produces a decrease of the tunneling energy when the wave functions in the two wells overlap to some extent. Especially for larger values of the nonlinear interaction term, results from this model produce better agreement with numerical solutions of the time-dependent Gross-Pitaevskii equation in one and three dimensions, as compared with models that have no interaction term in the tunneling energy. The usefulness of this model is demonstrated by good agreement with recent experimental results for the tunneling oscillation frequency [Albiez et al., Phys. Rev. Lett. 95, 010402 (2005)]. We also present equations and results for a multimode approach, and use the I2M model to obtain modified equations for the second-quantized version of the Bose-Einstein double-well problem.
TL;DR: In this paper, the authors consider a set of atoms trapped in an isotropic harmonic potential, with infinite scattering length, and obtain exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2,1) hidden symmetry, existence of a decoupled bosonic degree of freedom, and relations between the moments of the trapping potential energy and the moment of the total energy.
Abstract: We consider $N$ atoms trapped in an isotropic harmonic potential, with $s$-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2,1) hidden symmetry, existence of a decoupled bosonic degree of freedom, and relations between the moments of the trapping potential energy and the moments of the total energy.