Time-frequency as quantum continuous variables

Abstract

We present a second quantization description of frequency-based continuous variables quantum computation in the subspace of single photons. For this, we define frequency and time operators using the free field Hamiltonian and its Fourier transform, and show that these observables, when restricted to the one photon per mode subspace, reproduce the canonical position-momentum commutation relations. As a consequence, frequency and time operators can be used to define a universal set of gates in this particular subspace. We discuss the physical implementation of these gates as well as their effect on single photon states, and show that frequency and time variables can also be used to implement continuous variables quantum information protocols, in the same way than polarization is currently used as a two-dimensional quantum variable.

Quantum optics with time-frequency degrees of freedom of single photons

“…In the single photon subspace, we have shown in Ref. 1 that the time and frequency operator do not commute, following Eq. ( 1):…”

“…However, the mathematical equivalent of the beam-splitter in the time-frequency domain for two single photons, referred to as the "frequency beam-splitter operation" that we have presented in Ref 1,3,18 has been approximatively implemented experimentally. Such an operation takes the form e i(ωj Tk +ω k Tj ) , and act on two single photon state as:…”

“…The source of the quantumness in time and frequency variables can be attributed to the non-commutativity of time and frequency operators, which can be defined properly in the one photon per mode subspace. As a result, the frequency and time operators can be utilized to establish a comprehensive set of gates in this specific subspace Ref 1 . The similarities and differences between the time-frequency and quadrature position-momentum degrees of freedom in both mathematical and physical terms are explained in Ref 1 .…”

“…As a result, the frequency and time operators can be utilized to establish a comprehensive set of gates in this specific subspace Ref 1 . The similarities and differences between the time-frequency and quadrature position-momentum degrees of freedom in both mathematical and physical terms are explained in Ref 1 . The quadrature position-momentum variables, which are quantum continuous variables can be the amplitude and phase of the electromagnetic field and are in this sense particle sensitive variables.…”

Reconstructing the full modal structure of photonic states by stimulated emission tomography

“…The resulting post-selected state becomes a valuable nongaussian resource in quantum optics and quantum information. This state can be, for instance, highly entangled in different modes, as polarization -that can be used to encode qubits -or frequency -that can be used to encode qubits, qudits [36,37], or for continuous variables for quantum computing [38][39][40] and metrology [41].…”