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Journal of Modern Optics

The barriers to producing multiparticle superposition states in rotating Bose–Einstein condensates

D. W. Hallwood, Keith R. Burnett, Jacob Dunningham

Published Sep 10, 2007• 23 citations
Publisher

Abstract

We study why it is quite so hard to make a superposition of superfluid flows in a Bose–Einstein condensate. To do this we initially investigate the quantum states of N atoms trapped in a 1D ring with a barrier at one position and a phase applied around it. We show how macroscopic superpositions can in principle be produced and investigate factors which affect the superposition. We then use the Bose–Hubbard model to study an array of Bose–Einstein condensates trapped in optical potentials and coupled to one another to form a ring. We derive analytic expressions for the quality of the superposition for this system, which agrees well with direct diagonalization of the Hamiltonian for relatively small numbers of atoms. We show that for macroscopic superpositions to be realized there are essentially three straightforward requirements, other than an absence of decoherence, which become harder to achieve as the system size increases. Firstly, the energies of the two distinct superfluid states must be sufficiently close. Secondly, coupling between the two states must be sufficiently strong, and thirdly, other states must be well separated from those participating in the superposition.

DOI
10.1080/09500340701427128

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•Dec 1, 2011•3 citations
We study why it is quite so hard to make a superposition of superfluid flows in a Bose–Einstein condensate. To do this we initially investigate the quantum states of N atoms trapped in a 1D ring with a barrier at one position and a phase applied around it. We show how macroscopic superpositions can in principle be produced and investigate factors which affect the superposition. We then use the Bose–Hubbard model to study an array of Bose–Einstein condensates trapped in optical potentials and coupled to one another to form a ring. We derive analytic expressions for the quality of the superposition for this system, which agrees well with direct diagonalization of the Hamiltonian for relatively small numbers of atoms. We show that for macroscopic superpositions to be realized there are essentially three straightforward requirements, other than an absence of decoherence, which become harder to achieve as the system size increases. Firstly, the energies of the two distinct superfluid states must be sufficiently close. Secondly, coupling between the two states must be sufficiently strong, and thirdly, other states must be well separated from those participating in the superposition.
We study why it is quite so hard to make a superposition of superfluid flows in a Bose-Einstein condensate. To do this we initially investigate the quantum states of NNN atoms trapped in a 1D ring with a barrier at one position and a phase applied around it. We show how macroscopic superpositions can in principle be produced and investigate factors which affect the superposition. We then use the Bose-Hubbard model to study an array of Bose-Einstein condensates trapped in optical potentials and coupled to one another to form a ring. We derive analytic expressions for the quality of the superposition for this system, which agrees well with direct diagonalisation of the Hamiltonian for relatively small numbers of atoms. We show that for macroscopic superpositions to be realised there are essentially three straightforward requirements, other than an absence of decoherence, which become harder to achieve as the system size increases. Firstly, the energies of the two distinct superfluid states must be sufficiently close. Secondly, coupling between the two states must be sufficiently strong, and thirdly, other states must be well separated from those participating in the superposition.
We present a scheme for creating macroscopic superpositions of the direction of superfluid flow around a loop. Using the Bose–Hubbard model, we study an array of Bose–Einstein condensates (BECs) trapped in optical potentials and coupled to one another to form a ring. By rotating the ring so that each particle acquires on average half a quantum of superfluid flow, it is possible to create a multiparticle superposition of all the particles rotating and all the particles stationary. Under certain conditions it is possible to scale up the number of particles to form a macroscopic superposition. The simplicity of the model has allowed us to study macroscopic superpositions at an atomic level for different variables. Here, we concentrate on the tunnelling strength between the potentials. Further investigation remains important, because it could lead us to making an ultra-precise quantum-limited gyroscope.
We present a straightforward scheme for creating macroscopic superpositions of different superfluid flow states of Bose-Einstein condensates trapped in optical lattices. This scheme has the great advantage that all the techniques required are within the reach of current experiments. Furthermore, the relative difficulty of creating superpositions scales favorably with the size of the state. This means that this scheme may be well suited to creating superpositions involving large numbers of particles. Such states may have interesting technological applications such as making quantum-limited measurements of angular momentum.
Quantum behaviour appears to be lost in macroscopic objects. They do not, for example, form superpositions. In this paper I present two schemes that produce multiparticle superpositions of superfluid flow. By studying the effect of external parameters on the superposition, I have found three conditions that must be satisfied for a superposition to form. The energy of the superposition states must be close, there must be a coupling path between the states and other states must not couple strongly. By optimising these conditions, it may be possible to make Schrödinger cat states, useful for metrology and stable qubits.
We give a theoretical analysis of the experimental scheme for creating macroscopic superpositions of different superfluid flow states with Bose-Einstein condensates trapped in an optical lattice [J. Dunningham et al., Phys. Rev. A 74, 023601 (2006)]. It is shown that a class of macroscopic superposition states can be created for an arbitrary number of particles by dynamical evolution with a properly prepared initial state. When the number of particles is a multiple of 3, such states are exactly reduced to macroscopic superpositions of different superfluid flow states.
Superfluidity is a remarkable phenomenon. Superfluidity was initially characterized by flow without friction, first seen in liquid helium in 1938, and has been studied extensively since. Superfluidity is believed to be related to, but not identical to Bose-Einstein condensation, a statistical mechanical phenomena predicted by Albert Einstein in 1924 based on the statistics of Satyendra Nath Bose, where bosonic atoms make a phase transition to form a Bose-Einstein condensate (BEC), a gas which has macroscopic occupation of a single quantum state. Developments in laser cooling of neutral atoms and the subsequent realization of Bose-Einstein condensates in ultracold gases have opened a new window into the study of superfluidity and its relation to Bose-Einstein condensation. In our atomic sodium BEC experiment, we studied superfluidity and dissipationless flow in an all-optical toroidal trap, constructed using the combination of a horizontal ``sheet''-like beam and vertical ``ring''-like beam, which, like a circuit loop, allows flow around the ring. On inducing a single quantum of circulation in the condensate, the smoothness and uniformity of the toroidal BEC enabled the sustaining of a persistent current lasting 40 seconds, limited by the lifetime of the BEC due to background gas pressure. This success set the stage for further experiments studying superfluidity. In a first set of experiments, we studied the stability of the persistent current by inserting a barrier in the flow path of the ring. The superflow stopped abruptly at a barrier strength such that the local flow velocity at the barrier exceeded a critical velocity, which supported decay via the creation of a vortex-antivortex pair. Our precise control in inducing and arresting superflow in the BEC is a first step toward studying other aspects of superfluidity, such as the effect of temperature and dimensionality. This thesis discusses these experiments and also details partial-transfer absorption imaging, an imaging technique developed in the course of this work.