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Applied Mathematics & Information Sciences

Schrodinger Cats in Double Well Bose Condensates: Modeling Their Collapse and Detection via Quantum State Diffusion

William P. Reinhardt, Cynthia A. Stanich, Cory D. Schillaci

Published Jan 1, 2009• 2 citations
Publisher

Abstract

Can macroscopic quantum superposition states (or highly entangled number states) be observed directly'? Specifically, can phase contrast imaging be applied to observe a superposition state with essentially "all" of the atoms in a gaseous double well BEG being simultaneously in both wells at the same time'? That is we are looking to image states of the type vertical bar N, 0 > +vertical bar 0, N > where vertical bar L, R > denotes L particles in the Left well and R. in the Right. We will happily settle for states of the form IN n, n > +In, N is >, with 71, << N, these being less ephemeral. Earlier work in our group, Perry, Reinhardt and Kahn, has shown that such highly entangled number states may be generated by appropriate phase engineering, just as in the case of the phase engineering of solitons in single well BECs. Experimentalists have been hesitant to attempt to create such states in fear that definitive observations cannot be carried out. There have also been suggestions that "Nature" will prevent such superpositions from existing for N too large... and thus there are also basic issues in quantum theory which may prevent the formation and detection of such states. In the present progress report we begin an investigation of calculating the lifetimes of such entangled states in the presence of both observation and spontaneous decay both of which perturb, and eventually destroy, the entanglement under investigation via quantum back-action. Quantum State Diffusion (QSD) provides a useful computational tool in addressing such questions, and we present the initial results of exploring this novel use of QSD.

DOI
10.12785/amis

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Can macroscopic quantum superposition states (or highly entangled number states) be observed directly'? Specifically, can phase contrast imaging be applied to observe a superposition state with essentially "all" of the atoms in a gaseous double well BEG being simultaneously in both wells at the same time'? That is we are looking to image states of the type vertical bar N, 0 > +vertical bar 0, N > where vertical bar L, R > denotes L particles in the Left well and R. in the Right. We will happily settle for states of the form IN n, n > +In, N is >, with 71, << N, these being less ephemeral. Earlier work in our group, Perry, Reinhardt and Kahn, has shown that such highly entangled number states may be generated by appropriate phase engineering, just as in the case of the phase engineering of solitons in single well BECs. Experimentalists have been hesitant to attempt to create such states in fear that definitive observations cannot be carried out. There have also been suggestions that "Nature" will prevent such superpositions from existing for N too large... and thus there are also basic issues in quantum theory which may prevent the formation and detection of such states. In the present progress report we begin an investigation of calculating the lifetimes of such entangled states in the presence of both observation and spontaneous decay both of which perturb, and eventually destroy, the entanglement under investigation via quantum back-action. Quantum State Diffusion (QSD) provides a useful computational tool in addressing such questions, and we present the initial results of exploring this novel use of QSD.
We provide an experimentally viable scheme for the generation of entangled number states of Bose–Einstein condensates in multiple wells with cyclic pairwise connectivity. The condensate ground state in a multiple well trap can self-evolve, when appropriately phase engineered with specific phase differences between the neighbouring wells, to a macroscopic superposition state with controllable entanglement – as for example the multiple well generalization of double well NOON states. We demonstrate, through numerical simulations, the creation of entangled states of small numbers of particles in three and four wells and then explore the creation of entangled states where there are larger numbers of particles or a larger number of wells. In the four well case we study a novel, initially unexpected, type of entanglement produced as the particle numbers, or interaction strengths, increase.
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.
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.
We study the possibility to create many-particle Schrödinger 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 experimental verification of coherence in the cat-like 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 cat-like state can survive multiple scatterings before collapse occurs. In addition, we study the dephasing effect of atom-atom collisions on the cat-like states.
We propose and numerically validate a scheme for the realization of Schrödinger cat-like states in a two-component Bose-Einstein condensate, emphasizing their twinning across components under tunable intra- and inter-species interactions within the miscibility regime. Wigner phase-space analysis reveals sub-Planck-scale interference fringes, confirming the nonclassical character of the states and their potential utility in quantum-enhanced metrology. The dynamical response of the system to a weak linear gravitational-like perturbation further demonstrates cooperative enhancement: while the directly perturbed component retains its cat-like features, the coupled partner exhibits a pronounced population imbalance and a distinct phase-space rotation, providing a sensitive detection channel absent in single-component condensates. These results establish binary condensates as a versatile platform for engineering macroscopic quantum superpositions and exploiting their twinning dynamics for precision measurements.