Abstract
We discuss a class of quantum theories which exhibit a sharply increased memory storage capacity due to emergent gapless degrees of freedom. Their realization, both theoretical and experimental, is of great interest. On the one hand, such systems are motivated from a quantum information point of view. On the other hand, they can provide a framework for simulating systems with enhanced capacity of pattern storage, such as black holes and neural networks. In this paper, we develop an analytic method that enables us to find critical states with increased storage capabilities in a generic system of cold bosons with weak attractive interactions. The enhancement of memory capacity arises when the occupation number N of certain modes reaches a critical level. Such modes, via negative energy couplings, assist others in becoming effectively gapless. This leads to degenerate microstates labeled by the occupation numbers of the nearly-gapless modes. In the limit of large N, they become exactly gapless and their decoherence time diverges. In this way, a system becomes an ideal storer of quantum information. We demonstrate our method on a prototype model of N attractive cold bosons contained in a one-dimensional box with Dirichlet boundary conditions. Although we limit ourselves to a truncated system, we observe a rich structure of quantum phases with a critical point of enhanced memory capacity.
Highlights
We discuss a class of quantum theories which exhibit a sharply increased memory storage capacity due to emergent gapless degrees of freedom
In the D-dimensional model of [4], the number of the emergent gapless modes and their microstate entropy scale as the area of a D – 1-dimensional sphere, in similarity to the black hole Bekenstein entropy. These results suggest that systems with emergent gapless modes can offer a pathway for understanding the microscopic origin of black hole entropy and holography on simple prototype systems
Before we can come to the main statement of this section, we introduce the notion of a critical point of the Bogoliubov Hamiltonian Hbog
Summary
We discuss a class of quantum theories which exhibit a sharply increased memory storage capacity due to emergent gapless degrees of freedom. Their realization, both theoretical and experimental, is of great interest. The enhancement of memory capacity arises when the occupation number N of certain modes reaches a critical level. Such modes, via negative energy couplings, assist others in becoming effectively gapless. The level of excitation of a mode k in a given state |nk can be conveniently described by an occupation number nk of the corresponding quantum oscillator, with the usual creation/annihilation operators a †k, ak and the number operator nk = a †kak This means that the transition between patterns in which the occupation numbers differ significantly is in general expected to be costly
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