Adaptive Homodyne Measurement

Michael Armen, John Au, Andrew Doherty, and John Stockton

Introduction

The merger of real-time feedback with quantum systems will provide an important practical ingredient to emerging technologies at the quantum scale. In addition, introducing feedback into a quantum setting provides an excellent paradigm for experiments in fundamental physics. Real-time feedback used on a quantum system (in contrast to that used on a classical system) must be designed to account for 'measurement back-action' and thus the performance of the feedback can provide a direct test of our quantitative description of the continuous evolution of measured quantum systems. Furthermore, feedback can be used to realize adaptive measurement protocols, which, in turn, allow us to realize a larger class of quantum measurements in the laboratory. Here in the Mabuchilab, we are interested in using feedback for quantum system control and for designing adaptive quantum measurements.

Background

The adaptive homodyne measurement is our lab's first experimental realization of an adaptive quantum measurement. Consider the metrological task of making a single-shot estimate of the phase (relative to a reference) of a weak optical pulse. This situation may come up in, for example, a coherent telecommunication scenario where information is encoded in the phase of each pulse in a pulse train. Since each pulse has a finite number of photons in it, quantum mechanics requires that the phase of that pulse have some intrinsic uncertainty to it. Therefore, an optimal estimate of the phase would be limited only by the intrinsic quantum uncertainty of the pulse itself. From a practical standpoint, no experimental measurement can ever exactly reach an intrinsic quantum limit, so we will refer to a measurement procedure as uncertainty-limited if the inherent measurement inaccuracy is small enough to be dominated by the intrinsic quantum uncertainty of the state.

The figure below shows a simple depiction of the situation considered above. A weak Signal pulse is combined with an intense Local Oscillator(LO) at a beamsplitter, and a detector collects the resulting interference signal. The Signal-LO relative phase can then be estimated by measuring the resulting intensity (which is phase dependent). However, the quantum fluctuations in the signal phase will lead to a 'noisy' interference signal (shown below), so that for an uncertainty-limited phase measurement, the Signal-LO relative phase should be set to 90 degrees (also called a phase-quadrature homodyne measurement). Unfortunately, a phase-quadrature homodyne measurement requires a priori knowledge of the signal phase, and thus cannot be considered a true phase measurement.

Prior to the work of H.M. Wiseman [1,2], there was no known measurement procedure that could even in principle provide an uncertainty-limited estimate of optical phase. The best practical measurement procedure known was heterodyne detection, which can at best come to within a factor of 2 of the quantum limit (for a heterodyne measurement, the LO phase is rapidly swept over 360 degrees, thereby sampling the interference fringe many times over the duration of the pulse). Wiseman was able to show that real-time feedback could be used to realize an uncertainty-limited optical phase measurement. As the leading edge of each pulse is measured, the resulting photocurrent is used to make an quick estimate of the signal phase. The LO can then be adjusted so as to be optimal for the remainder of that pulse, thereby realizing an adaptive homodyne measurement. A careful analysis predicts that this adaptive homodyne measurement is essentially uncertainty-limited for signals with an average photon number ~10 (or greater).

The Experiment

From an experimental viewpoint, the adaptive homodyne measurement is convenient while providing some unique technical challenges. It is convenient in the sense that the measurement requires only linear optical elements (beamsplitters, phase-shifters, etc...) and electronic feedback. Below is a schematic of our experiment [3,4]. Light from a stabilized Nd:Yag laser enters a Mach-Zehnder interferometer (MZI), creating two beams with a well defined relative phase. The LO is generated in an acousto-optic modulator (AOM), while the signal pulse is the RF sideband created in an electro-optic modulator (EOM). The LO and signal interfere at the 50/50 beamsplitter, and the resulting signals are collected by two photodetectors. The adaptive feedback algorithm is implemented by a field programmable gate array (FPGA) [5] that processes the signal photocurrent and quickly outputs a phase adjustment. The adaptive measurement is realized by adjusting the phase of the RF driving the EOM (which adjusts the signal phase). To see some typical electronic traces of an adaptive measurement, click here. In addition, we perform heterodyne measurements in this same apparatus by simply detuning the RF frequency driving the EOM and turning off the FPGA feedback.

In addition to the components listed above, the experiment relies on a frequency multiplexing scheme that allows us to generate an interferometric signal for stabilizing the length of the MZI. A small fraction of the LO interferes with the EOM carrier, and the resulting heterodyne signal (collected by a separate pair of detectors) is used to stabilize the MZI via a feedback loop and a piezo electric stack (PZT). This feedback loop has nothing to do with the adaptive measurement procedure, it is only used to allow for long periods of data acquisition.

The two main technical challenges are the need for broadband quantum-noise limited measurement sensitivity and high-speed digital signal processing. To achieve quantum-noise limited sensitivity both the optical source and the detectors must have minimal extraneous noise. Two of the major obstacles for the success of the adaptive experiment were the elimination of relaxation-oscillation noise on the Nd:Yag laser (via a Fabry-Perot cavity) and the design of broadband shot-noise limited photodetectors.

With a view towards developing the technological know-how for future experiments in real-time quantum feedback, we used an FPGA to serve our digital signal processing needs. The FPGA is comprised of an array of re-programmable logic gates whose functions can be specified using VHDL code. The rapid prototyping and design flexibility associated with FPGAs makes them convenient tools for experimental quantum optics.

The FPGA board displayed on the left is equipped with 4 connectorized inputs/outputs, each with its own 12 bit ADC/DAC. There are two FPGA chips (the large square components near the center of the board) that clock at 100 MHz and each contain ~3 million system gates (>10,000 logic cells). For virtually all algorithms we have implemented with this board, the performance has been limited mainly by the group delay of ~150 nsec imposed by the input/output anti-aliasing filters. (The board pictured is the GVA-290 purchased from GV & Associates).

Results

We perform heterodyne and adaptive homodyne measurements on 50usec signal pulses containing ~1-1000 photons per pulse. In the range of ~10 - 300 photons per pulse, the adaptive phase measurement is superior to heterodyne detection, and the adaptive phase estimate variances lie closer to the pulse intrinsic quantum limit. Above this range, extraneous noise introduced by the electronic feedback loop limits the adaptive measurement performance. Below ~10 photons per pulse the adaptive measurement is limited by low signal-to-noise.

The data plotted on the left is the phase estimate variance vs. pulse photon number. The intrinsic phase variance for each pulse is plotted as a thick solid line. In the range of 10-300 photons per pulse, most of the adaptive data lies below the curve for theoretically perfect heterodyne detection (dotted line), and all of it lies below the heterodyne curve that includes the effects of electronic noise (solid line). The excellent agreement between the heterodyne data and the solid line verifies that our state preparation and signal processing procedures are sound.

Current Work

We are currently investigating the main sources of limitation for the adaptive measurement. Although we are not planning on continuing the adaptive homodyne experiment long into the future, understanding the limitations for this experiment will prove useful for future investigation involving quantum limited measurement and feedback.

(last updated 29 February 2003)

References

Group publications