Towards the Ideal Quantum Measurement
Author: Juergen Volz
Affiliation: Laboratoire Kastler Brossel de l'E.N.S., Paris, France
Measurement lies at the heart of quantum physics and gives rise to many of the counter-intuitive aspects of the theory. In a classical world, a measurement can in principle be performed with arbitrary precision without disturbing the system. In contrast, a quantum mechanical measurement inevitably projects the system into one of its basis states, although initially the system may have been in a superposition of these states. For example, it is possible to prepare an atom in a superposition of two internal states. However, a state measurement will always yield the result that the atom is either in one or the other state, which is typically referred to as 'collapse of the wave function'. This collapse during the measurement process constitutes the unavoidable back-action of the measurement on the measured system and gives rise to phenomena as for example Heisenberg's uncertainty relation.
While theoretically well described, experimental realizations typically fall short of these predictions, always causing a back-action on the system being measured -- with orders of magnitude larger than required for an ideal measurement. This additional back-action typically results in a energy transfer to the quantum system and heating which is a major drawback for many experiments.
The modern field of quantum optics and quantum information, relies on the accurate readout of quantum information stored in so-called quantum bits, i.e. single quantum objects that for example can be realized using the internal states of single atoms or ions. The most efficient detection method for internal atomic states is the fluorescence detection. Here an atom with two stable ground states is subject to an incident light field resonant to a transition of one of the two ground states to an excited state. If the atom is in the resonant state, it will be repeatedly excited and - after spontaneously emitting a photon - decay back into the original state, while an atom in the off-resonant state is not affected by light field. The presence (or absence) of fluorescence photons then indicates the atomic state. However, this requires the atom to undergo a large amount of spontaneous emissions events, leading to an inevitable energy exchange between atom and light field.
Fig 1: (a) Simplified level scheme of Rubidium. (b) Principle of the resonator measurement where the transmission properties of the resonator yield information on the atomic state (all light reflected: atom is in the resonant state, all light transmitted: atom is in the off-resonant state). (c) Schematic view of the experimental setup with the fiber-resonator implemented on an atomchip.
Theoretically, an ideal atomic state measurement only projects the atom onto its basis states and requires no spontaneous scattering. Therefore, our research team at the Ecole Normale Supérieure in Paris decided to investigate if we can reach a regime where we can perform a measurement with significantly less than one spontaneous emission event. In our experiments we use as measurement device a so-called Fabry-Perot resonator which consists of two highly reflecting mirrors facing each other. This allows to keep light inside the resonator for approximately 38000 round trips before it is lost. With the help of a novel fiber-optical technology, the mirrors are directly imprinted on the tips of two optical fibers. In this way we can produce miniaturized resonators with a small enough volume so that a single atom placed between its mirrors is enough to shift the resonance frequency by a sizeable amount. As long as the atom is in the off resonant state it does not affect the resonator and a laser tuned to the empty cavity’s resonance is fully transmitted through the cavity. If, however, the atom is in the resonant state the resonance frequency changes and nearly all of the incident light is reflected without ever entering the resonator, thereby avoiding spontaneous scattering.
Fig 2: Experimental results for the detection process. The blue data points are the residual error of the atomic state measurement, plotted as function of average number of scattered photons. The green curve is the minimum error possible for our resonator and the grey area corresponds to the regime accessible without resonator.
To investigate the exact amount of residual scattering in our experiment, we determine the state of the atom after each measurement from which we can deduce that our measurement allows us to infer the atomic state with a fidelity of more than 90% while scattering only 0.2 photons on average. The main limit of our measurement scheme is the small probability to finally detect the incident photons. In order to measure how our system would perform under ideal circumstances, we also directly analyzed the fundamental measurement back-action on the atom using the quantum Zeno effect. This effect states that permanently measuring a physical system will stop its temporal evolution and freeze it in its current state. In our experiment, we apply a microwave pulse to transfer the atomic state to the other. At the same time we perform our state detection which permanently projects the atom into its initial state an thereby prevents the transfer. This allows us to directly measure the projection rate of the atom into its basis states, from which we conclude that three photons incident on the cavity are enough for a full collapse of the atomic state.
These results demonstrate that during the measurement (nearly) no spontaneous emission occurred and the back-action on the atom approaches the fundamental limit given by the uncertainty principle. Besides giving a compelling illustration of the quantum measurement process, these results have important consequences for quantum information applications with atoms or ions, allowing internal state readout without any heating, hereby allowing much higher cycling rates. In addition, this new detection scheme promises the development of efficient optical detectors for complex quantum systems as e.g. single molecules.
 Maximilian Schlosshauer, "Decoherence, the measurement problem, and interpretations of quantum mechanics", Rev. Mod. Phys. 76, 1267 (2005). Abstract.
 A. H. Myerson, et al., "High-Fidelity Readout of Trapped-Ion Qubits", Phys. Rev. Lett 100, 200502 (2008). Abstract.
 Jürgen Volz, Roger Gehr, Guilhem Dubois, Jérôme Estève and Jakob Reichel, "Measurement of the internal state of a single atom without energy exchange", Nature 475, 210 (2011). Abstract.
 Wayne M. Itano, "Perspectives on the quantum Zeno paradox.", arXiv:quant-ph/0612187v1 (2006).