We have modulated the density of a trapped Bose-Einstein condensate by changing the trap stiffness, thereby modulating the speed of sound. We observe the creation of correlated excitations with equal and opposite momenta, and show that for a well-defined modulation frequency, the frequency of the excitations is half that of the trap modulation frequency.

Quantum mechanics is a very successful and still intriguing theory, introducing two major counter-intuitive concepts. Wave-particle duality means that objects normally described as particles, such as electrons, can also behave as waves, while entities primarily described as waves, such as light, can also behave as particles.This revolutionary idea nevertheless relies on notions borrowed from classical physics, either waves or particles evolving in our ordinary space-time. By contrast, entanglement leads to interferences between the amplitudes of multi-particle states, which happen in an abstract mathematical space and have no classical counterpart. This fundamental feature has been strikingly demonstrated by the violation of Bell's inequalities [1][2][3][4] . There is, however, a conceptually simpler situation in which the interference between two-particle amplitudes entails a behaviour impossible to describe by any classical model. It was realised in the Hong, Ou and Mandel (HOM) experiment 5 , in which two photons arriving simultaneously in the input channels of a beam-splitter always emerge together in one of the output channels. In this letter, we report on the realisation, with atoms, of a HOM experiment closely following the original protocol. This opens the prospect of testing Bell's inequalities involving mechanical observables of massive particles, such as momentum, using methods inspired by quantum optics 6,7 , with an eye on theories of the quantum-to-classical transition [8][9][10][11] . Our work also demonstrates a new way to produce and benchmark twin-atom pairs 12,13 that may be of interest for quantum information processing 14 and quantum simulation 15 . 1 arXiv:1501.03065v2 [quant-ph] 15 Jan 2015A pair of entangled particles is described by a state vector that cannot be factored as a product of two state vectors associated with each particle. Although entanglement does not require that the two particles be identical 2 , it arises naturally in systems of indistinguishable particles due to the symmetrisation of the state. A remarkable illustration is the HOM experiment, in which two photons enter in the two input channels of a beam-splitter and one measures the correlation between the signals produced by photon counters placed at the two output channels. A joint detection at these detectors arises from two possible processes: either both photons are transmitted by the beam-splitter or both are reflected (Fig. 1c). If the two photons are indistinguishable, both processes lead to the same final quantum state and the probability of joint detection results from the addition of their amplitudes. Because of elementary properties of the beam-splitter, these amplitudes have same modulus but opposite signs, thus their sum vanishes and so also the probability of joint detection (Refs. [16,17] and Methods). In fact, to be fully indistinguishable, not only must the photons have the same energy and polarisation, but their final spatio-temporal modes must be identical. In the HOM experiment, it means that the ...

The Cauchy-Schwarz (CS) inequality -one of the most widely used and important inequalities in mathematics -can be formulated as an upper bound to the strength of correlations between classically fluctuating quantities. Quantum-mechanical correlations can, however, exceed classical bounds. Here we realize four-wave mixing of atomic matter waves using colliding Bose-Einstein condensates, and demonstrate the violation of a multimode CS inequality for atom number correlations in opposite zones of the collision halo. The correlated atoms have large spatial separations and therefore open new opportunities for extending fundamental quantum-nonlocality tests to ensembles of massive particles. 03.75.Gg, 34.50.Cx, 42.50.Dv The Cauchy-Schwarz (CS) inequality is ubiquitous in mathematics and physics [1]. Its utility ranges from proofs of basic theorems in linear algebra to the derivation of the Heisenberg uncertainty principle. In its basic form, the CS inequality simply states that the absolute value of the inner product of two vectors cannot be larger than the product of their lengths. In probability theory and classical physics the CS inequality can be applied to fluctuating quantities and states that the expectation value of the cross-correlation I 1 I 2 between two quantities I 1 and I 2 is bounded from above by the auto-correlations in each quantity:This inequality is satisfied, for example, by two classical currents emanating from a common source. In quantum mechanics, correlations can, however, be stronger than those allowed by the CS inequality [2][3][4]. Such correlations have been demonstrated in quantum optics using, for example, antibunched photons produced via spontaneous emission [5], or twin photon beams generated in a radiative cascade [6], parametric down conversion [7], and optical fourwave mixing [8]. Here the discrete nature of the light and the strong correlation (or anticorrelation in antibunching) between photons is responsible for the violation of the CS inequality. The violation has even been demonstrated for two light beams detected as continuous variables [8].In this work we demonstrate a violation of the CS inequality in matter-wave optics using pair-correlated atoms formed in a collision of two Bose-Einstein condensates (BECs) of metastable helium [9-12] (see Fig. 1). The CS inequality which we study is a multimode inequality, involving integrated atomic densities, and therefore is different from the typical two-mode situation studied in quantum optics. Our results demonstrate the potential of atom optics experiments to extend the fundamental tests of quantum mechanics to ensem- Spherical halo of scattered atoms produced by four-wave mixing after the cloud expands and the atoms fall to the detector 46 cm below. During the flight to the detector, the unscattered condensates acquire a disk shape shown in white on the north and south poles of the halo. The (red) boxes 1 and 2 illustrate a pair of diametrically symmetric counting zones (integration volumes) for the average cross-and autocorrela...

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In many-body systems governed by pairwise contact interactions, a wide range of observables is linked by a single parameter, the two-body contact, which quantifies two-particle correlations. This profound insight has transformed our understanding of strongly interacting Fermi gases. Here, using Ramsey interferometry, we study coherent evolution of the resonantly interacting Bose gas, and show that it cannot be explained by only pairwise correlations. Our experiments reveal the crucial role of three-body correlations arising from Efimov physics, and provide a direct measurement of the associated three-body contact.A fundamental challenge in many-body quantum physics is to connect the macroscopic behaviour of a system to the microscopic interactions between its constituents. In ultracold atomic gases the strength of interactions is most commonly characterised by the s-wave scattering length a, which can be tuned via Feshbach resonances [1]. On resonance a diverges and one reaches the unitary regime, in which the interactions are as strong as allowed by quantum mechanics. This regime has been extensively studied in Fermi gases [2][3][4], while the unitary Bose gas represents a new experimental frontier [5][6][7][8][9][10].In these systems, universal properties of the short-range particle correlations imply universal thermodynamic relations between macroscopic observables such as the momentum distribution, energy, and the spectroscopic response [11][12][13][14][15][16][17][18][19]. In the case of (mass-balanced) two-component Fermi gases, at the heart of these relations is a single fundamental thermodynamic parameter, the two-body contact density C 2 , which measures the strength of two-particle correlations. However, the case of the Bose gas is more subtle. In this system Efimov physics gives rise to three-body bound states [20][21][22][23][24][25][26], and more generally introduces three-particle correlations that cannot be deduced from the knowledge of pairwise ones [17][18][19]27]. The implication for many-body physics is that complete understanding of the macroscopic coherent phenomena requires knowledge of both C 2 and its three-body analogue C 3 [17][18][19].The relative importance of three-particle correlations generally grows with the strength of interactions. At moderate interaction strengths C 2 was measured spectroscopically, but C 3 was not observed [24]. However, the momentum distribution of the unitary Bose gas [7] suggested deviations from two-body physics [19,28].Here we interferometrically measure both C 2 and C 3 in a resonantly interacting thermal Bose gas, and find excellent agreement with theoretical predictions. The idea of our experiment is illustrated in Fig. 1. We perform radio-frequency (RF) Ramsey interferometry on a gas of atoms with two internal (spin) states, ↑ and ↓, and use a magnetic Feshbach resonance to enhance ↑↑ interactions, while both ↑↓ and ↓↓ interactions are negligible. For a measurement at a given magneticRamsey interferometry of a many-body system. The first π/2 pulse pu...

We measure the quantum depletion of an interacting homogeneous Bose-Einstein condensate and confirm the 70-year-old theory of Bogoliubov. The observed condensate depletion is reversibly tunable by changing the strength of the interparticle interactions. Our atomic homogeneous condensate is produced in an optical-box trap, the interactions are tuned via a magnetic Feshbach resonance, and the condensed fraction is determined by momentum-selective two-photon Bragg scattering.

We study the dynamics of an initially degenerate homogeneous Bose gas after an interaction quench to the unitary regime at a magnetic Feshbach resonance. As the cloud decays and heats, it exhibits a crossover from degenerate-to thermal-gas behaviour, both of which are characterised by universal scaling laws linking the particle-loss rate to the total atom number N . In the degenerate and thermal regimes the per-particle loss rate is ∝ N 2/3 and N 26/9 , respectively. The crossover occurs at a universal kinetic energy per particle and at a universal time after the quench, in units of energy and time set by the gas density. By slowly sweeping the magnetic field away from the resonance and creating a mixture of atoms and molecules, we also map out the dynamics of correlations in the unitary gas, which display a universal temporal scaling with the gas density, and reach a steady state while the gas is still degenerate.Strong interactions and correlations are at the heart of the most interesting many-body quantum phenomena. The possibility to control the interaction strength via Feshbach resonances [1] makes ultracold atomic gases an excellent setting for studies of strongly correlated behaviour. On resonance, the s-wave scattering length a, which characterises two-body contact interactions, diverges. In this so-called unitary regime the interactions are as strong as allowed by quantum mechanics, and the physics cannot explicitly depend on a, leading to the possibility of new types of universal behaviour [2][3][4][5][6].Of particular interest are the interaction-dominated degenerate unitary gases. Within the 'universality hypothesis', they have only one relevant lengthscale -the average interparticle spacing, given by the density n, which also sets the natural energy and time scales [2]:where m is the atom mass. These 'Fermi energy' and 'Fermi time' scales are applicable to both Fermi and Bose gases. In Bose gases, however, the universality can be broken by Efimov physics [7][8][9][10][11][12][13][14][15][16][17]. The Feshbach dimer molecular state, responsible for the resonance, is of size a and becomes unbound as a → ∞, but the infinite series of Efimov trimer states, each of a size 22.7 times larger than the previous one, can introduce new lengthscales into the problem. While unitary Fermi gases have been extensively explored [3][4][5], the experimental [15,16,[18][19][20][21] and theoretical [22][23][24][25][26][27][28][29][30][31][32][33][34] studies of unitary Bose gases are only recently emerging. An experimental challenge is that they exhibit rapid threebody loss and heating, which also raises fundamental questions about the extent to which they have well defined equilibrium properties [20], but the loss dynamics also offer a valuable probe of the unitary behaviour [16,[18][19][20][21]. While coherent three-body correlations [15] and Efimov trimers [16] have been observed, the decay dynamics [16,[18][19][20][21] have been consistent with universal scalings (see also [35]). All experiments so far were performe...

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