Abstract
In 1924 David Hilbert conceived a paradoxical tale involving a hotel with an infinite number of rooms to illustrate some aspects of the mathematical notion of "infinity." In continuous-variable quantum mechanics we routinely make use of infinite state spaces: here we show that such a theoretical apparatus can accommodate an analog of Hilbert's hotel paradox. We devise a protocol that, mimicking what happens to the guests of the hotel, maps the amplitudes of an infinite eigenbasis to twice their original quantum number in a coherent and deterministic manner, producing infinitely many unoccupied levels in the process. We demonstrate the feasibility of the protocol by experimentally realizing it on the orbital angular momentum of a paraxial field. This new non-Gaussian operation may be exploited, for example, for enhancing the sensitivity of NOON states, for increasing the capacity of a channel, or for multiplexing multiple channels into a single one.
Highlights
In 1924 David Hilbert conceived a paradoxical tale involving a hotel with an infinite number of rooms to illustrate some aspects of the mathematical notion of “infinity.” In continuous-variable quantum mechanics we routinely make use of infinite state spaces: here we show that such a theoretical apparatus can accommodate an analog of Hilbert’s hotel paradox
We demonstrate the feasibility of the protocol by experimentally realizing it on the orbital angular momentum of a paraxial field
We first show how to map the eigenstate amplitudes of an infinite square potential well to twice their original level, and we report results of a physical implementation of an analogous protocol on the orbital angular momentum (OAM) eigenstates of light, where we coherently multiply any linear superposition by a fixed integer
Summary
In 1924 David Hilbert conceived a paradoxical tale involving a hotel with an infinite number of rooms to illustrate some aspects of the mathematical notion of “infinity.” In continuous-variable quantum mechanics we routinely make use of infinite state spaces: here we show that such a theoretical apparatus can accommodate an analog of Hilbert’s hotel paradox. We first show how to map the eigenstate amplitudes of an infinite square potential well to twice their original level, and we report results of a physical implementation of an analogous protocol on the orbital angular momentum (OAM) eigenstates of light, where we coherently multiply any linear superposition by a fixed integer (in our case, by three).
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