Separable Schmidt modes of a nonseparable state

Abstract

Two-photon states entangled in continuous variables such as wavevector or frequency represent a powerful resource for quantum information protocols in higher-dimensional Hilbert spaces. At the same time, there is a problem of addressing separately the corresponding Schmidt modes. We propose a method of engineering two-photon spectral amplitude in such a way that it contains several non-overlapping Schmidt modes, each of which can be filtered losslessly. The method is based on spontaneous parametric down-conversion (SPDC) pumped by radiation with a comb-like spectrum. There are many ways of producing such a spectrum; here we consider the simplest one, namely passing the pump beam through a Fabry-Perot interferometer. For the two-photon spectral amplitude (TPSA) to consist of non-overlapping Schmidt modes, the crystal dispersion dependence, the length of the crystal, the Fabry-Perot free spectral range and its finesse should satisfy certain conditions. We experimentally demonstrate the control of TPSA through these parameters. We also discuss a possibility to realize a similar situation using cavity-based SPDC.PACS numbers: 42.50. Dv, 42.50.Ar, 03.65.Ta, 85.60.Gz Introduction.Entanglement of two-photon states (biphotons) in continuous variables such as frequency or wavevector suggests the use of biphotons as a quantum-information resource in higher-dimensional Hilbert spaces [1,2]. The dimensionality of the Hilbert space is determined in this case by the Schmidt number, the effective number of Schmidt modes, which can reach several hundred [2][3][4]. But in order to realize any protocols with multimode states, it is necessary to address single Schmidt modes separately. For wavevector variables, any single Schmidt mode can be filtered out using a single-mode fibre and a spatial light modulator [6]. This filtering, in principle, can be lossless, which is crucial for experiments with twin-beam squeezing [7][8][9][10][11][12]. For frequency variables, it is far more difficult to losslessly select a single Schmidt mode. Attempts are being made in homodyne detection [13]; in direct detection experiments the procedure is more difficult. For instance, methods based on nonlinear frequency conversion are proposed, technically complicated to realize with high efficiency [14]. Here we propose and demonstrate a method of engineering a two-photon state in such a way that it contains non-overlapping Schmidt modes, each of which can be filtered losslessly using a spectral device.Frequency entanglement of biphotons. A two-photon state generated via SPDC can be written in the formwhereâ † s ,â † i are the creation operators of the signal and idler photons, and F (ω s , ω i ) represents the two-photon spectral amplitude (TPSA) [15]. The TPSA fully characterizes the spectral properties of a biphoton state and its physical meaning is the joint spectral probability amplitude of the down-converted photons in signal and idler modes with frequencies ω s and ω i respectively. TPSA is used to determine the degree of frequency entanglemen...

Separated Schmidt modes in the angular spectrum of biphotons

“…3b). Similarly to how it was demonstrated in a frequency domain [7], it can be realized for spatial mode in the near or far field ( Fig. 4).…”

“…To reflect this fact we inserted a parameter α = 0.63 into Eq. (7) and took it into account in all theoretical calculations:…”

Spectral properties near the Mott transition in the two-dimensional t-J model with next-nearest-neighbor hopping

“…1(b), 1(c), and 1(e)] can be identified 40 as the giant kink and waterfall behavior observed in high-T c cuprates 7-11 for which various interpretations have been proposed. [7][8][9][10][11]20,21,40,72,73,[99][100][101][102] Modes ii ′ and iii ′ correspond to spinon-like and holon-like branches observed in high-T c cuprates, 7 respectively.…”

“…The properties around (π, 0) are primarily characterized by mode vi ′ , which corresponds to the flat band observed in high-T c cuprates and in theoretical calculations. 5,12,13,40,[63][64][65][66]71,73,[93][94][95][96][97]102,103 Because this mode carries large spectral weights and exhibits an almost flat dispersion relation over a wide momentum range around (π, 0), it significantly contributes to the main peak of the single-particle density of states ρ(ω) [Figs. 1(a)-1(c)].…”

Coherence properties of high-gain twin beams

“…At low pump powers, where no filters were used to attenuate the light of twin beams, the numbers K are nearly identical. We note that the dual basis revealed in the Schmidt decomposition [38,39] can successfully be replaced for more intense fields by the input-output eigenmodes of the Bloch-Messiah reduction of the signal-idler unitary evolution operators [36].…”