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 Hilberts 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 realising 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 N00N states, for increasing the capacity of a channel or for multiplexing multiple channels into a single one.
Retrieving the vast amount of information carried by a photon is an enduring challenge in quantum metrology science and quantum photonics research. The transverse spatial state of a photon is a convenient high-dimensional quantum system for study, as
it has a well-understood classical analogue as the transverse complex field profile of an optical beam. One severe drawback of all currently available quantum metrology techniques is the need for a time-consuming characterization process, which scales very unfavorably with the dimensionality of the quantum system. Here we demonstrate a technique that directly measures a million-dimensional photonic spatial state in a single setting. Through the arrangement of a weak measurement of momentum and parallel strong measurements of position, the complex values of the entire photon state vector become measurable directly. The dimension of our measured state is approximately four orders of magnitude larger than previously measured. Our work opens up a practical route for characterizing high-dimensional quantum systems in real time. Furthermore, our demonstration also serve as a high-speed, extremely-high-resolution unambiguous complex field measurement technique for diverse classical applications.
Quantum key distributions (QKD) systems often rely on polarization of light for encoding, thus limiting the amount of information that can be sent per photon and placing tight bounds on the error that such a system can tolerate. Here we describe a pr
oof-of-principle experiment that indicates the feasibility of high-dimensional QKD based on the transverse structure of the light field, allowing for the transfer of more than 1 bit per photon. Our implementation uses the orbital angular momentum (OAM) of photons and the corresponding mutually unbiased basis of angular position (ANG). Our experiment uses a digital micro-mirror device for the rapid generation of OAM and ANG modes at 4 kHz, and a mode sorter capable of sorting single photons based on their OAM and ANG content with a separation efficiency of 93%. Through the use of a 7-dimensional alphabet encoded in the OAM and ANG bases, we achieve a channel capacity of 2.05 bits per sifted photon. Our experiment shows that, in addition to having an increased information capacity, QKD systems based on spatial-mode encoding will be more tolerant to errors and thus more robust against eavesdropping attacks.
We describe an experimental technique to generate a quasi-monochromatic field with any arbitrary spatial coherence properties that can be described by the cross-spectral density function, $W(mathbf{r_1,r_2})$. This is done by using a dynamic binary a
mplitude grating generated by a digital micromirror device (DMD) to rapidly alternate between a set of coherent fields, creating an incoherent mix of modes that represent the coherent mode decomposition of the desired $W(mathbf{r_1,r_2})$. This method was then demonstrated experimentally by interfering two plane waves and then spatially varying the coherent between these two modes such that the interference fringe visibility was shown to vary spatially between the two beams in an arbitrary and prescribed way.
We present a weak measurement protocol that permits a sensitive estimation of angular rotations based on the concept of weak-value amplification. The shift in the state of a pointer, in both angular position and the conjugate orbital angular momentum
bases, is used to estimate angular rotations. This is done by an amplification of both the real and imaginary parts of the weak-value of a polarization operator that has been coupled to the pointer, which is a spatial mode, via a spin-orbit coupling. Our experiment demonstrates the first realization of weak-value amplification in the azimuthal degree of freedom. We have achieved effective amplification factors as large as 100, providing a sensitivity that is on par with more complicated methods that employ quantum states of light or extremely large values of orbital angular momentum.
Orbital angular momentum (OAM) of light is an attractive degree of freedom for funda- mentals studies in quantum mechanics. In addition, the discrete unbounded state-space of OAM has been used to enhance classical and quantum communications. Unambigu
ous mea- surement of OAM is a key part of all such experiments. However, state-of-the-art methods for separating single photons carrying a large number of different OAM values are limited to a theoretical separation efficiency of about 77 percent. Here we demonstrate a method which uses a series of unitary optical transformations to enable the measurement of lights OAM with an experimental separation efficiency of more than 92 percent. Further, we demonstrate the separation of modes in the angular position basis, which is mutually unbiased with respect to the OAM basis. The high degree of certainty achieved by our method makes it particu- larly attractive for enhancing the information capacity of multi-level quantum cryptography systems.
Free-space communication allows one to use spatial mode encoding, which is susceptible to the effects of diffraction and turbulence. Here, we discuss the optimum communication modes of a system while taking such effects into account. We construct a f
ree-space communication system that encodes information onto the plane-wave (PW) modes of light. We study the performance of this system in the presence of atmospheric turbulence, and compare it with previous results for a system employing orbital-angular-momentum (OAM) encoding. We are able to show that the PW basis is the preferred basis set for communication through atmospheric turbulence for a large Fresnel number system. This study has important implications for high-dimensional quantum key distribution systems.
We describe a procedure by which a long ($gtrsim 1,mathrm{km}$) optical path through atmospheric turbulence can be experimentally simulated in a controlled fashion and scaled down to distances easily accessible in a laboratory setting. This procedure
is then used to simulate a 1-km-long free-space communication link in which information is encoded in orbital angular momentum (OAM) spatial modes. We also demonstrate that standard adaptive optics methods can be used to mitigate many of the effects of thick atmospheric turbulence.
