Quantum noise in three-dimensional BEC interferometry

Abstract

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. PACS numbers: 03.75. Gg, 03.75.Dg, 67.85.Fg, 67.85.De Atom interferometry is an important quantum technology at the heart of many proposed future applications of ultra-cold atomic physics. Bose-Einstein condensates (BECs) or atom lasers are macroscopic quantum objects and have potential advantages as interferometric detectors and sensors, provided one can precisely extract atomic phase information. However, unlike photons, atoms can interact strongly, causing dephasing and loss of interference fringes. An intimate understanding of quantum many-body dynamics is the key to calculating interaction-induced dephasing in the measurement process. This is essential for a quantitative theory of atom interferometry.In this Letter we present a simple, yet quantitatively accurate theoretical approach to simulating the dynamics and evaluating limits of atom interferometry at large atom number, using a truncated Wigner representation [1][2][3]. This method extends the conventional Gross-Pitaevskii equations describing a Bose condensate to include quantum noise effects, including noise due to linear and nonlinear losses. The theory allows the accurate inclusion of quantum fluctuations due to nonlinear losses, which is a dominant effect when atom numbers are increased to improve fringe visibility.Importantly, we can clearly demonstrate where fringe visibility is driven by quantum fluctuations, and where it is driven by trap inhomogeneity and dynamical effects, in order to choose optimal conditions for quantum noise reduction and spin squeezing. These calculations are a first step towards understanding mesoscopic superpositions and entanglement in ultra-cold atomic gases. An advantage of our method compared to the variational approaches used elsewhere [4,5] is that it allows us to treat a large number of independent field modes and particles, thus including degrees of freedom that are excited due to collisional and nonlinear loss dynamics [6,7]. Our theory can be readily extended to include finite temperature initial conditions [2,8], which will be treated elsewhere. Nonlinear losses and finite temperature effects can be also described within the confines of the variational approach [9,10].Quantum phase-diffusion is defined as the phase noise induced by number fluctuations which are conjugate to phase. This is a fundamental feature of BEC interferometry, and can only be removed when there are no interactions. However, there are other reasons for decoherence, which are also important. The approach used here captures all three significant features of atom interferometry that can result in decoherence: phase-diffusion, losses, and trap inhomogeneity effe...

Spontaneous spin squeezing in a rubidium BEC

“…To estimate the effective nonlinearity for the different choices of the scattering lengths, we calculated the parameter χ of the one-axis twisting Hamiltonian at the stationary state in our geometry. We obtain χ=7.5×10 −5 s 1 with the scattering length values from [32] (red curve), χ=7.3×10 −5 s 1 with the values from [20] (green curve), and 24.6 10 s 5 1 c =´-for the combination [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] (blue curve).…”

“…Furthermore, as we have seen, the asymmetric losses significantly affect the state of the system over the relatively long times needed for the spontaneous spin squeezing to occur. In order to take into account all of these features in a consistent way, we performed simulations using a Wigner method inspired by [35]. To limit the drawbacks of the truncated Wigner method [36]-which are related to the fact that the added quantum noise in each mode efficiently thermalizes in 3D, introducing spurious effects-we implemented a 'minimal version' of the Wigner method, where we project the quantum noise on the condensate mode for each component. )…”

Einstein-Podolsky-Rosen-entangled Bose-Einstein condensates in state-dependent potentials: A dynamical study

“…To complete our investigation, we have evaluated the total loss rate in the interaction configuration (middle row in Fig.1) using stationary wave functions for N a = N b = 500, scattering lengths and trap frequencies as in Fig.2, and the loss rate constants [27] κ 11 = 81 × 10 −21 m 3 s −1 , κ 01 = 15 × 10 −21 m 3 s −1 for twobody losses and κ 000 = 5.4 × 10 −42 m 6 s −1 for three-body losses. We find that about 7 particles are lost on average in 0.2 s due to two-body losses, while three-body losses are negligible.…”

Quantum-field-theoretical approach to phase–space techniques: Symmetric Wick theorem and multitime Wigner representation

“…Midgley et al performed a comparison of the truncated Wigner with the exact positive-P representation and a Hartree-Fock-Bogoliubov approximate method for the simulation of molecular BEC dissociation [20], concluding that the truncated Wigner representation was the most useful in practical terms. This practical usefulness has been demonstrated in studies of BEC interferometry [21], domain formation in inhomogeneous ferromagnetic dipolar condensates [22], vortex unbinding following a quantum quench [23], a reverse Kibble-Zurek mechanism in two-dimensional superfluids [24], the quantum and thermal effects of dark solitons in one-dimensional Bose gases [25], the quantum dynamics of multi-well Bose-Hubbard models [26,27], and analysis of a method to produce Einstein-Podolsky-Rosen states in two-well BEC [28]. Along with its continuing use in quantum optics, the above examples demonstrate that the truncated Wigner representation is an extremely useful approximation method, allowing for the numerical simulation of a number of processes for which nothing else is known to be as effective.…”