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Demonstrating Superpositions in Double-Well Bose-Einstein Condensates

Tania J. Haigh

Published May 1, 2010• 0 citations
OpenAlex

Abstract

This thesis considers measurements that could be used to verify the existence of a coherent superposition, using the specific example of a superposition of a Bose-Einstein condensate in a double-well. There is significant interest in superposition states in atomic condensates, due to the possibility of large numbers of atoms being involved in the superposition, and while various experimental proposals exist for generating such states none have yet been successfully verified in this system. We use a number-state description of the condensate in a double well, which is a number-conserving approach with basis states specifying the number of atoms in each well. This allows us to consider 'NOON`-type states, i.e. a superposition of the 'N,0' state of all atoms localized in the left well and the '0,N' state where all N atoms are localized in the right hand well. MATLAB is used to solve for the ground state, dynamics, and measurement expectation values for the double-well condensate in this basis. We consider measurements that would distinguish this coherent superposition from a statistical mixture of the same states. Such a suitable measurement is the parity of the atoms in one well, after a Ramsey-type interference experiment has been performed. We have found that unambiguously distinguishing even a perfect 'NOON' state from a statistical mixture using standard detection methods will present experimental difficulties. We find that interactions between condensate atoms degrade the visibility of the parity oscillations, and we also consider the effects of detector inefficiency on these parity measurements. We suggest that a 'NOON'-type superposition state in the double-well system could be generated in a reasonable time frame by making use of a Feshbach resonance to tune the inter-atomic interactions from repulsive to negative. We find that even with imperfect superpositions, parity measurements retain their frequency dependence, which is the signature of NOON-type coherence.

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This thesis considers measurements that could be used to verify the existence of a coherent superposition, using the specific example of a superposition of a Bose-Einstein condensate in a double-well. There is significant interest in superposition states in atomic condensates, due to the possibility of large numbers of atoms being involved in the superposition, and while various experimental proposals exist for generating such states none have yet been successfully verified in this system. We use a number-state description of the condensate in a double well, which is a number-conserving approach with basis states specifying the number of atoms in each well. This allows us to consider 'NOON`-type states, i.e. a superposition of the 'N,0' state of all atoms localized in the left well and the '0,N' state where all N atoms are localized in the right hand well. MATLAB is used to solve for the ground state, dynamics, and measurement expectation values for the double-well condensate in this basis. We consider measurements that would distinguish this coherent superposition from a statistical mixture of the same states. Such a suitable measurement is the parity of the atoms in one well, after a Ramsey-type interference experiment has been performed. We have found that unambiguously distinguishing even a perfect 'NOON' state from a statistical mixture using standard detection methods will present experimental difficulties. We find that interactions between condensate atoms degrade the visibility of the parity oscillations, and we also consider the effects of detector inefficiency on these parity measurements. We suggest that a 'NOON'-type superposition state in the double-well system could be generated in a reasonable time frame by making use of a Feshbach resonance to tune the inter-atomic interactions from repulsive to negative. We find that even with imperfect superpositions, parity measurements retain their frequency dependence, which is the signature of NOON-type coherence.
Quantum superpositions of macroscopically distinguishable states having distinct phases can be created with a Bose-Einstein condensate trapped in a periodic potential. The experimental signature is contained in the phase distribution of the interference patterns obtained after releasing the traps. Moreover, in the double-well case, this distribution exhibits a dramatic dependence on the parity of the total number of atoms. We finally show that, for single-well occupations up to a few hundred atoms, the macroscopic quantum superposition can be robust enough against decoherence to be experimentally revealable within current technology.
We create atom-molecule superpositions in a Bose-Fermi mixture of 87Rb^{87}\text{R}\text{b}87Rb and 40K^{40}\text{K}40K atoms. The superpositions are generated by ramping an applied magnetic field near an interspecies Fano-Feshbach resonance to coherently couple atom and molecule states. Rabi- and Ramsey-type experiments show oscillations in the molecule population that persist as long as $150\text{ }\ensuremath{\mu}\text{s}$ and have up to 50% contrast. The frequencies of these oscillations are magnetic-field dependent and consistent with the predicted molecule binding energy. This quantum superposition involves a molecule and a pair of free particles with different statistics (i.e., bosons and fermions) and furthers exploration of atom-molecule coherence in systems without a Bose-Einstein condensate.
Ultracold atoms and Bose-Einstein condensates represent a burgeoning and ever exciting area of research which has produced numerous breakthroughs in the last 15 years. Ultracold atoms at nano-Kelvin temperatures, isolated in atom traps from the hot, room temperature environment, exhibit ultimate quantum properties and can serve as pure samples for testing quantum mechanics and performing precision measurements. The research described in this thesis investigates, both theoretically and experimentally, the coherence and dynamical evolution of a two-component Bose-Einstein condensate (BEC). The studies are performed on the |1 › ≡ F = 1,mF = −1 › and |2 › ≡ |F = 2,mF = +1 › hyperfine ground states of a 87Rb BEC trapped on an atom chip. Conventional understanding has been that the fringe contrast in interferometric experiments on a BEC is limited by the mean-field dephasing due to strong interatomic interactions which lead to inhomogeneous collisional shifts. We have discovered using Ramsey interferometry a mean-field driven self-rephasing effect in a trapped twocomponent BEC. When combined with a spin-echo technique, we find that the selfrephasing leads to a coherence time of 2.8 s, the longest ever recorded for an interacting BEC. Secondly, we have developed a new technique based on periodic collective oscillations for precision measurements of the interspecies and intraspecies scattering lengths and derived an analytic mean-field theory for the phase and density dynamics in the twocomponent BEC. This technique has been applied to the measurements of the scattering lengths for the two components, |1i and |2i, in 87Rb with a precision of 0.016%. Additionally, the two-body loss coefficients for these states have been measured. Thirdly, we have developed and applied a new, interferometric method for calibrating the detection system, in order to determine the total atom number precisely for Ramsey interferometric measurements. The calibration coefficient is found to be in a good agreement with that obtained using a conventional calibration technique based on the critical BEC temperature. Finally, we have detected predicted RF-induced Feshbach resonances by monitoring changes in the two-body loss coefficients in the two-component BEC. Conventional Feshbach resonances have been used in many works to tune the s-wave interactions; however, there is no achievable magnetic Feshbach resonance for magnetically trappable states of 87Rb. The RF-induced resonances which we have detected can provide a way to tune the scattering lengths which is important for interferometry and entanglement experiments. The positions of the resonances may provide useful information for predicting the atomic scattering properties of 87Rb.
In recent years, atom interferometry has become established as an indispensable tool in both fundamental and applied physics. With present state-of-the-art devices based on thermal atoms reaching limits imposed by the momentum spread of the initial atomic wavepacket, it seems natural to ask whether colder sources such as Bose-Einstein condensates may prove beneficial in advancing the precision of interferometric measurements. The thesis at hand aims to inform this question, specifically by examining the role played by atomic interactions in interferometers based on Bose-condensed atoms. Interactions can have both advantageous and deleterious consequences in the context of atom interferometry. They provide a means to control the momentum width of the condensate, and facilitate the generation of nonclassical squeezed states which may enhance the phase sensitivity beyond the shot noise limit. Conversely, the condensate self-interaction causes mean-field shifts, multimode excitations and phase diffusion which can erode both the precision and the accuracy of an interferometric measurement. The question of when and in which systems the detrimental effects of interactions outweigh the advantages of using Bose-Einstein condensates is an important one, and warrants investigation. This thesis presents experimental studies into the role of interactions in both internal- and external-state atom interferometers. As a foundation for these investigations, we describe the design and construction of an apparatus for creating Bose-Einstein condensates of the two stable rubidium isotopes in an optical trap. By sympathetic cooling with a rubidium-87 reservoir, we are able to produce condensates of rubidium-85 in which the interactions may be adjusted by means of a magnetic Feshbach resonance. The tunability afforded by the Feshbach resonance is used to study inelastic losses in ultracold rubidium-85 clouds, as well as the effect of interactions on condensate stability and on the ground state of dual-species mixtures. In particular, we offer new experimental data on the dynamics of collapsing condensates with attractive interactions, over which some controversy has existed since the first experiments more than a decade ago. Good agreement is found between the measured collapse times and a simple mean-field model. Proceeding to interferometry, we present results from Ramsey interferometers operating on the clock transition of rubidium-87 Bose-Einstein condensates. In free-space operation with Raman beamsplitters, we demonstrate projection-noise-limited performance, an important prerequisite for the realisation of squeezing-enhanced sensitivity. Using large condensates of up to 10⁶ atoms and microwave coupling, we study the effect of interactions on the Ramsey fringe contrast. The dominant source of decoherence is found to be spatial dynamics driven by the difference in interparticle interaction strengths, which are analysed using the spin-echo technique and numerical simulations of the Gross-Pitaevskii equation. Finally, we turn our attention to external-state interferometry, implementing a Mach-Zehnder gravimeter using Bragg transitions in a freely falling rubidium-87 condensate. Large-momentum-transfer beamsplitters composed of higher-order Bragg diffraction and Bloch oscillations are used to increase the accumulated phase and thus the sensitivity of the interferometer. The role of interactions in this system is examined, and we canvass methods for achieving further increases in sensitivity.
We study the possibility of creating many-particle macroscopic quantum superposition (Schr\"odinger cat)--like states by using a Feshbach resonance to reverse the sign of the scattering length of a Bose-Einstein condensate trapped in a double-well potential. To address the issue of the experimental verification of coherence in the catlike state, we study the revival of the initial condensate state in the presence of environmentally induced decoherence. As a source of decoherence, we consider the interaction between the atoms and the electromagnetic vacuum, due to the polarization induced by an incident laser field. We find that the resulting decoherence is directly related to the rate at which spontaneously scattered photons carry away sufficient information to distinguish between the two atom distributions which make-up the cat state. We show that for a ``perfect'' cat state, a single scattered photon will bring about a collapse of the superposition, while a less-than-perfect catlike state can survive multiple scatterings before the collapse occurs. In addition, we study the dephasing effect of atom-atom collisions on the catlike states.
Summary form only given. We present a potential scheme to detect weak forces by employing a BEC confined in a double-well potential. Initially the condensate is prepared in a coherent superposition of having all the atoms in one well or the other. We allow the condensate to experience a phase shift for a certain time /spl tau/ under the influence of the force. The phase shift can be detected by using the technique analogous to the Ramsey interference. The interference fringes can be read out by performing a homodyne measurement on the condensate localized in one well. We have analyzed the limitations on the accuracy of the scheme and show that a high-precision measurement can be achieved if the condensate contains a large number of coherently condensed atoms.