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arXiv:1407.6389 · Physical Review A

Multimode quantum state tomography using unbalanced array detection

Brandon S. Harms, Blake E. Anthony, Noah T. Holte, Hunter A. Dassonville, Andrew M. C. Dawes

Published Nov 10, 2014• 2 citations
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Abstract

We measure the joint $Q$ function of a multispatial-mode field using a charge-coupled device array detector in an unbalanced heterodyne configuration. The intensity pattern formed by interference between a weak signal field and a strong local oscillator is resolved using Fourier analysis and used to reconstruct quadrature amplitude statistics for 22 spatial modes simultaneously. The local oscillator and signal propagate at an angle of $12\phantom{\rule{0.28em}{0ex}}\mathrm{mrad}$, thus shifting the classical noise to modes that do not overlap with the signal. In this configuration, balanced detection is not necessary.

DOI
10.1103/physreva.90.053818

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We measure the joint QQQ function of a multispatial-mode field using a charge-coupled device array detector in an unbalanced heterodyne configuration. The intensity pattern formed by interference between a weak signal field and a strong local oscillator is resolved using Fourier analysis and used to reconstruct quadrature amplitude statistics for 22 spatial modes simultaneously. The local oscillator and signal propagate at an angle of 12mrad12\phantom{\rule{0.28em}{0ex}}\mathrm{mrad}12mrad, thus shifting the classical noise to modes that do not overlap with the signal. In this configuration, balanced detection is not necessary.
We have measured the joint Q-function of a highly multimode field using unbalanced heterodyne detection with a charge-coupled device array detector. We use spectral interferometry between a weak signal field and a strong, 100 fs duration local oscillator pulse to reconstruct the joint quadrature amplitude statistics of about 25 temporal modes. By adjusting the time delay between the signal and local oscillator pulses we are able to shift all the classical noise to modes distinct from the signal. This obviates the need to use a balanced detector.
We demonstrate that it is possible to use the balanced homodyning with array detectors to measure the quantum state of a correlated two-mode signal field. We show the applicability of the method to fields with complex mode functions, thus generalizing the work of Beck [Phys. Rev. Lett. 84, 5748 (2000)] in several important ways. We further establish that, under suitable conditions, array detector measurements from one of the two ouputs are sufficient to determine the quantum state of signal. We show the power of the method by reconstructing a truncated Perelomov state, which exhibits complicated structure in the joint probability density for the quadratures.
I propose a method for measuring the quantum state of an optical field that occupies a mode having a complicated spatial structure. The technique uses array detectors and a single, plane-wave local oscillator beam. The advantage of using array detectors is that the local oscillator is not mode matched to the field being measured, yet the deleterious effects of this mismatch on the effective detection efficiency are greatly reduced compared to using single detectors. Indeed, when the spatial mode of the signal field is describable by a real function, the effective mode-matching efficiency is unity.
Phase-controlled homodyne 2(N+R)-port detection with R local oscillators is analyzed with the aim of reconstructing the quantum state of a correlated N-mode signal field. It is shown that both the (N+R)-fold joint count distributions and the N-fold joint difference-count distributions contain all knowable information on the state of the field. In any case, the minimum number of local oscillators is given by the number of different signal-field frequencies. Two Fourier integrals per mode are found to be required to reconstruct the density matrix of the signal field from the joint difference-count distributions measured in balanced homodyning. To illustrate the theory, it is applied to balanced homodyne 4N-port detection (R=N). In this scheme, the joint difference-count distributions directly yield the joint field-strength distributions of a correlated N-mode signal field.
We have simultaneously measured the quantum states of two different spatial modes of the same optical beam by performing quantum-state tomography with an array detector. Both modes are well described by coherent states, but the projection of the signal onto the local oscillator mode contains a mean of 0.09 photons, while a more complicated mode has a mean of 4.3 photons. This demonstrates that for this particular mode the effective detection efficiency when using array detection is over 40 times greater than when using single detectors.