We investigate the atom-optical analog of degenerate four-wave mixing by colliding two Bose-Einstein condensates of metastable helium. The momentum distribution of the scattered atoms is measured in three dimensions. A simple analogy with photon phase matching conditions suggests a spherical final distribution. We find, however, that it is an ellipsoid with radii smaller than the initial collision momenta. Numerical and analytical calculations agree with this and reveal the interplay between many-body effects, mean-field interaction, and the anisotropy of the source condensate. The field of atom optics has developed to the point that one can now speak of the beginning of ''quantum-atom optics'' [1] in which atoms are manipulated in ways similar to photons and in which quantum fluctuations and entanglement play an important role. The demonstration of atom pair production [2,3], either from the dissociation of ultracold molecules, a process analogous to parametric downconversion [4-6], or from collisions of Bose-Einstein condensates (BECs) [7][8][9][10], analogous to four-wave mixing (FWM) [11][12][13][14][15][16][17][18][19][20][21], holds considerable promise for generating atomic squeezed states and demonstrating nonlocal Einstein-Podolsky-Rosen (EPR) correlations [4,5,22,23]. In both these systems, atom-atom interactions play the role of the nonlinear medium that allows conversion processes. Atoms are not, however, exactly like photons, and in spite of their formal similarity, the processes of pair production of photons and of atoms exhibit some interesting and even surprising differences that must be understood in order for the quantum-atom optics field to advance. In this work, we discuss one such effect.In optical FWM or parametric down-conversion [24], energy conservation requires that the sum of the energies of the outgoing photons be fixed by the energy of the input photon(s). Phase matching requirements impose constraints on the directions and values of the individual photon momenta. A simple case is degenerate, spontaneous FWM (i.e., two input photons of equal energy) in an isotropic medium, for which energy conservation and phase matching require that the momenta of the output photons lie on a spherical shell whose radius is that of the momenta of the input photons.We have performed the atom-optical analog of degenerate FWM in colliding BECs while paying careful attention to the momenta of the outgoing atoms. We find that unlike the optical case, the output momenta do not lie on a sphere, but rather on an ellipsoid with short radius smaller than the input momentum. This behavior is due to a subtle combination of atom-atom interactions, which impose an energy cost for pair production, and the anisotropy of the condensates, which affects the scattered atoms as they leave the interaction region.Although an analogous effect could exist in optics, optical nonlinearities are typically so small that the effect is negligible. However, in the process of high-harmonic generation in intense laser fields, a simil...
We study the beyond-mean-field corrections to the energy of a dipolar Bose gas confined to two dimensions by a box potential with dipoles oriented in plane. At a critical strength of the dipolar interaction the system becomes unstable on the mean field level. We find that the ground state of the gas is strongly influenced by the corrections, leading to formation of a self-bound droplet, in analogy to the free space case. Properties of the droplet state can be found by minimizing the extended Gross-Pitaevskii energy functional. In the limit of strong confinement we show analytically that the correction can be interpreted as an effective three-body repulsion which stabilizes the gas at finite density. arXiv:1911.02384v1 [cond-mat.quant-gas]
We formulate the time-dependent Bogoliubov dynamics of colliding Bose-Einstein condensates in terms of a positive-P representation of the Bogoliubov field. We obtain stochastic evolution equations for the field which converge to the full Bogoliubov description as the number of realisations grows. The numerical effort grows linearly with the size of the computational lattice. We benchmark the efficiency and accuracy of our description against Wigner distribution and exact positive-P methods. We consider its regime of applicability, and show that it is the most efficient method in the common situation -when the total particle number in the system is insufficient for a truncated Wigner treatment.
We consider a sonic analog of a black hole realized in the one-dimensional flow of a Bose-Einstein condensate. Our theoretical analysis demonstrates that one- and two-body momentum distributions accessible by present-day experimental techniques provide clear direct evidence (i) of the occurrence of a sonic horizon, (ii) of the associated acoustic Hawking radiation, and (iii) of the quantum nature of the Hawking process. The signature of the quantum behavior persists even at temperatures larger than the chemical potential.
We apply an analytical model for anisotropic, colliding Bose-Einstein condensates in a spontaneous four wave mixing geometry to evaluate the second order correlation function of the field of scattered atoms. Our approach uses quantized scattering modes and the equivalent of a classical, undepleted pump approximation. Results to lowest order in perturbation theory are compared with a recent experiment and with other theoretical approaches.
Bragg diffraction divides a Bose-Einstein condensate into two overlapping components, moving with respect to each other with high momentum. Elastic collisions between atoms from distinct wave packets can significantly deplete the condensate. Recently, Ziń et al. ͓Phys. Rev. Lett. 94, 200401 ͑2005͔͒ introduced a model of two counterpropagating atomic Gaussian wave packets incorporating the dynamics of the incoherent scattering processes. Here we study the properties of this model in detail, including the nature of the transition from spontaneous to stimulated scattering. Within the first-order approximation, we derive analytical expressions for the density matrix and anomalous density that provide excellent insight into correlation properties of scattered atoms.
We define a 2 ϫ 2 matrix of Green's functions, G͑z,k͒ ª ͫ G 11 ͑z,k͒ G 21 ͑z,k͒ G 12 ͑z,k͒ G 22 ͑z,k͒ ͬ ,Note that, using the notation of Appendix H,We use the conventions described in this appendix for the meaning of Green's functions both away from the real line and on the real line.It is a general fact, which does not depend on the details of the system, thatBy the reflection invariance of the Bose gas, G ij ͑z,k͒ = G ij ͑z,− k͒. ͑5.25͒Obviously, for any observable A, ͗A ء ͘ = ͗A͘. But the state ͗ · ͘ is real, hence ͗A͘ = ͗Ā ͘. Note also that H = H , a k = ā k . Therefore, G 12 ͑z,k͒ = G 21 ͑z,k͒.Note that G 11 ͑0,k͒ = ͗͗a k ء , a k ͘͘ and G 12 ͑0,k͒ = ͗͗a k , a −k ͘͘, hence by ͑5.22͒,Let us introduce the "full mass operator,"⌺͑z,k͒ = ͫ ⌺ 11 ͑z,k͒ ⌺ 12 ͑z,k͒ ⌺ 21 ͑z,k͒ ⌺ 22 ͑z,k͒ ͬ ª 1 2 ͫ G 11 ͑z,k͒ G 12 ͑z,k͒ G 21 ͑z,k͒ G 22 ͑z,k͒ ͬ −1 = 1 2 ͑G 11 ͑z,k͒G 22 ͑z,k͒ − G 12 ͑z,k͒G 21 ͑z,k͒͒ −1 ͫ G 22 ͑z,k͒ − G 12 ͑z,k͒ − G 21 ͑z,k͒ G 11 ͑z,k͒ ͬ . Consequently,⌺ 11 ͑0,k͒ − ⌺ 12 ͑0,k͒ = 1 2͑G 11 ͑0,k͒ − G 12 ͑0,k͒͒ .͑5.26͒ implies 062103-38 Cornean, Dereziński, and Ziń
Mean field approximation treats only coherent aspects of the evolution of a Bose-Einstein condensate. However, in many experiments some atoms scatter out of the condensate. We study a semianalytic model of two counterpropagating atomic Gaussian wave packets incorporating the dynamics of incoherent scattering processes. Within the model we can treat processes of the elastic collision of atoms into the initially empty modes, and observe how, with growing occupation, the bosonic enhancement is slowly kicking in. A condition for the bosonic enhancement effect is found in terms of relevant parameters. Scattered atoms form a squeezed state. Not only are we able to calculate the dynamics of mode occupation, but also the full statistics of scattered atoms.
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