Quantum Hilbert Hotel

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.

Figure 1

Hilbert hotel protocol.—(a) The initial state is a single particle wave function $\psi (x)$ within an infinite square potential well. (b) We instantaneously expand the well to twice its original width. The original wave function is not immediately changed but the eigenbasis is different. (c) We allow free evolution for a period corresponding to the original fundamental period. The wave function is reflected around the center of the expanded well, with an undesired phase shift. (d) We insert an infinite barrier in the center (where the wave function is zero) to split it into two independent wells that evolve separately, an energy shift on one well corrects the relative phase. (e) After the phase correction we align the potentials and merge the two halves back together. (f) An adiabatic compression of the well maps the eigenstates of the expanded well to those of the original well. The original wave function has now been halved and reflected, corresponding to the Hilbert hotel operation ${\widehat{\mathbb{H}}}_2$ being applied to the eigenstates $\psi (x)$.

Figure 2

Experimental schematic.—A HeNe laser beam, spatially filtered with a single mode fiber (SMF) and collimated, is directed onto a phase-only spatial light modulator (SLM-1) to generate the desired combination of input OAM eigenmodes. The beam is then sent through a pair of machined polymer refractive elements that comprise the first OAM sorter. The optical field at the output plane of this sorter is imaged onto the top half of a second SLM implementing a fan-out grating. The fan-out was set to produce three copies of the beam, resulting in a $\times 3$ multiplication of the OAM quantum number of each mode. The Fourier plane of this grating is imaged onto the bottom half of the same SLM displaying the appropriate hologram to correct for the relative phase between the three copies. The three copies are then demagnified by a cylindrical lens and injected through a second OAM sorter operated in reverse. We measured the OAM spectrum of light at the output of the multiplier using a series of projective measurements for various values of $\ell$, which were implemented using a third SLM and a single-mode-fiber-coupled avalanche photodiode (APD).

Figure 3

Coherent OAM multiplication.—Top row: Near field of input coherent superpositions. Bottom row: Tripled output states. The number of petals is $6|\ell |$, as expected from a coherent operation.

Figure 4

OAM multiplication performance.—For each input eigenmode we measure the composition of the multiplied output. Circle size is linearly proportional to the overlap with the output modes. As can be seen, the small leakage onto the neighboring output modes is contained within a few adjacent modes. A sufficiently distant input superposition such as $|3⟩+|-3⟩$ would maintain an effective orthogonality.