We study the Josephson effect between atomic BoseEinstein condensates. By drawing on an electrostatic analogy, we derive a semiclassical functional expression for the three-dimensional Josephson coupling energy in terms of the condensate density. Estimates of the capacitive energy and of the Josephson plasma frequency are also given. The effect of dissipation due to the incoherent exchange of normal atoms is analysed. We conclude that coherent Josephson dynamics may already be observable in current experimental systems.Recently [1][2][3], it has become possible to cool a macroscopic number (∼ 10 4 − 10 6 ) of magnetically confined, spin-polarized atoms down to temperatures of order 100 nK while maintaining densities sufficiently high (10 11 − 10 15 cm −3 ) to permit the onset of Bose-Einstein condensation (BEC). From the ensuing theoretical work, it has been concluded that these Bose condensed atomic gases behave very differently from the ideal noninteracting gases, which yields the prospect of potentially displaying a rich phenomenology that might include vortex states and the Josephson effect [4][5][6].In this article we present a theoretical study of the Josephson dynamics between two atomic baths that have undergone BEC. The Josephson effect results from a collective mode of two weakly connected systems between which a macroscopic fraction of particles can tunnel with identical probability amplitude. Going beyond previously proposed one-dimensional [7] and dissipation free [7,8] models, we present here a three-dimensional study of the Josephson effect between Bose condensates and estimate the effect of damping. We calculate the Josephson coupling energy, the capacitive energy (which accounts for quantum fluctuations of the phase), and the frequency of the Josephson plasma oscillation. Our main conclusion is that still lower temperatures than those achieved up to date are needed for a clear realization of the Josephson effect.The collective dynamics of a Bose condensate at zero temperature is described by its macroscopic wave function Ψ(r, t). If this is factorized as Ψ = √ ρ exp(iϕ), the standard energy functional can be written as(g = 2πh 2 a/m), and the corresponding Hamilton equations lead to the Gross-Pitaveskii equations [5,9]. Neglecting depletion [10], the normalization can be taken as drρ = N , being N the total number of atoms.At sufficiently low temperatures the phase within one well can be regarded as uniform. This can be easily seen if one estimates the energy of a one-radian fluctuation of the phase across the condensate near equilibrium in a single spherical harmonic well [see Eq.(1)],where R = a 0 (15aN/a 0 ) 1/5 is the cloud radius estimated within the Thomas-Fermi approximation [4], ω 0 is the harmonic oscillator frequency within the well, and a 0 = (h/mω 0 ) 1/2 is the oscillator length. For the last approximate equality we have used typical parameters a = 5 nm and a 0 = 10 −4 cm. The characteristic temperature of such an oscillation can be as big as 10 µK for the MIT (1996) experiment [3]....

We argue that the phase across an asymmetric dc SQUID threaded by a magnetic flux can experience an effective ratchet (periodic and asymmetric) potential. Under an external ac current, a rocking ratchet mechanism operates whereby one sign of the time derivative of the phase is favored. We show that there exists a range of parameters in which a fixed sign (and, in a narrower range, even a fixed value) of the average voltage across the ring occurs, regardless of the sign of the external current dc component.PACS numbers: 05.40.+j, 74.40.+k, 74.50.+r, 85.25.Cp, 85.25.Dq Although the nonequibrium dynamics of a particle in a ratchet potential (i.e., a periodic potential that lacks reflection symmetry) has for long been considered a fundamental problem in statistical physics [1], it has become the object of more intense attention in recent years, because of its newly found relevance in diverse areas of physics, chemistry, and biology. A characteristic effect is that, when the ratchet is subject to a stationary nonequilibrium perturbation, particle motion in one direction is favored. Within this context, an important class of dynamical systems is formed by the so-called rocking ratchets, for which the external perturbation is a time periodic, uniform force [2][3][4]. The effect of dynamically induced unidirectional motion can overcome the drift effect of a small bias that would push the particle into the non favored direction. Thus, for not very strong tilts, uphill movement is possible provided the ratchet structure is conveniently rocked.In this letter, we propose a realization of the rocking ratchet mechanism in a new type of superconducting quantum interference device (SQUID) containing a characteristic asymmetry. The system we propose, depicted in Fig. 1, is formed by a ring with two Josephson junctions in series in one of the arms and only one junction in the other arm. We will show that, when the ring is threaded by a flux Φ ext that is not an integer multiple of Φ 0 /2 (Φ 0 ≡ h/2e being the flux quantum), the effective potential experienced by the total phase ϕ across the ring displays a ratchet structure. As a consequence, when the asymmetric SQUID is 'rocked' by an external ac current I(t), one sign of the phase velocityφ is favored. From the Josephson voltage-phase relation, we conclude that there must be a range of parameters for which a fixed sign of the average voltage V 0 ≡h φ /2e occurs regardless of the sign of the external current dc component I 0 .We focus on SQUID structures formed by conventional Josephson junctions whose phase is a classical variable and which can be adequately described by the 'resistively shunted junction' model [5,6]. Thus, the phase ϕ i across Josephson junction i on the left arm (see Fig. 1a) obeys the equation (i = 1, 2):where I l (t) is the current through the left arm, and R i , C i , and J i are the resistance, capacitance, and critical current of junction i. For simplicity, we assume here that the two junctions in series are identical, and will comment later on the cas...

We study double-barrier interfaces separating regions of asymptotically subsonic and supersonic flow of Bose condensed atoms. These setups contain at least one black hole sonic horizon from which the analog of Hawking radiation should be generated and emitted against the flow in the subsonic region. Multiple coherent scattering by the double-barrier structure strongly modulates the transmission probability of phonons, rendering it very sensitive to their frequency. As a result, resonant tunneling occurs with high probability within a few narrow frequency intervals. This gives rise to highly non-thermal spectra with sharp peaks. We find that these peaks are mostly associated to decaying resonances and only occasionally to dynamical instabilities. Even at achievable nonzero temperatures, the radiation peaks can be dominated by the spontaneous emission, i.e. enhanced zero-point fluctuations, and not, as often in analog models, by stimulated emission. PACS numbers: 03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow 04.62.+v Quantum fields in curved spacetime 04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics

The violation of a classical Cauchy-Schwarz (CS) inequality is identified as an unequivocal signature of spontaneous Hawking radiation in sonic black holes. This violation can be particularly large near the peaks in the radiation spectrum emitted from a resonant boson structure forming a sonic horizon. As a function of the frequency-dependent Hawking radiation intensity, we analyze the degree of CS violation and the maximum violation temperature for a double barrier structure separating two regions of subsonic and supersonic condensate flow. We also consider the case where the resonant sonic horizon is produced by a space-dependent contact interaction. In some cases, CS violation can be observed by direct atom counting in a time-of-flight experiment. We show that near the conventional zero-frequency peak, the decisive CS violation cannot occur.

We study the dynamics of the relative phase following the connection of two independently formed Bose-Einstein condensates. Dissipation is assumed to be due to the creation of quasiparticles induced by a fluctuating condensate particle number. The coherence between different values of the phase, which is characteristic of the initial Fock state, is quickly lost after the net exchange of a few atoms has taken place. This process effectively measures the phase and marks the onset of a semiclassical regime in which the system undergoes Bloch oscillations around the initial particle number. These fast oscillations excite quasiparticles within each condensate and the system relaxes at a longer time scale until it displays low-energy, damped Josephson plasma oscillations, eventually coming to a halt when the equilibrium configuration is finally reached.