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Quantum Nondemolition Measurements

In the field of quantum mechanics, quantum nondemolition (QND) refers to a measurement of a system without introducing back action onto the observables of interest. The Heisenberg uncertainty principle is well known for demonstrating a minimum amount of uncertainty between two observables. This holds true as long as the operators for both do not commute. A key feature in QND is taking advantage of an interaction in which the operators do commute and subsequently measuring the effect on the observables that are otherwise not of interest. This limits the back action introduced into the system onto the measured quantities without extending it to those of interest. In this manner, the desired observables may be indirectly detected without perturbation, potentially even multiple times.

History

Ideas of how one might design a QND experiment were introduced by Braginsky, Vorontsov, Thorne, Unruh, and Caves in the 1970s. However, The first examples of QND experiments were not demonstrated until 1995. While originally intended for study of gravitational waves, it has since shown a special amount of promise and usefulness in the field of quantum optics.

Example

QND is frequently demonstrated as a method of detecting and counting photons without absorbing them. Photons from a wide variety of frequencies have been non-destructively detected. Gerhard Rempe demonstrated QND using a maser. Later, he and Stephan Welte did so with optical photons. For an example of how this works, one can examine the work of Rempe and Welte.

Welte designed an experiment in which light was to be coupled into an optical fiber which ran through two QND detectors and ended at a Hanbury Brown-Twiss interferometer. Each QND detector consisted of an optical cavity with highly reflective mirrors, in which a single atom of rubidium-87 was trapped. The rubidium was optically pumped into an excited state for a 780-nm transition. Then, a pair of Raman lasers were used to create a π/2-pulse and induce a superposition of excited and ground states in the atoms.

Photons were then inserted into the optical fiber as a weak laser pulse. A photon interacting with the rubidium prepared thus would reflect off the atom and flip the atom's state. Note that this does still preserve the superposition of the atom. Thereafter, the atoms underwent another Raman π/2-pulse. At this point, if the atom had interacted with a photon, it would be restored to the excited state, whereas an atom that had not interacted would be moved into the ground state. The state of each atom was measured by allowing it to fluoresce and de-excite if it were excited.

Each pulse contained a controlled mean number of photons. The mean number would typically be kept at less than one, but had been tested from near zero to more than three. The Hanbury Brown-Twiss Interferometer at the end was used to determine the photon correlation function g(2)(τ), which could confirm the number of photons that had passed through in each pulse.

Welte found by this experiment that the same photon could be detected multiple times by consecutive QND detectors, definitively confirming the feasibility of quantum nondemolition.

Theory

As per the Heisenberg uncertainty principle, two non-commuting operators A, B exhibit a behavior in uncertainty as

$$\Delta A\Delta B\geq\frac{1}{2}[A,B]$$

where [A,B] denotes the commutator between the two. In order not to perturb the particle of interest, one mustn't perform any action on it involving operators such as these. Hence, in order to maintain nondemolition, the operators in play must be those that commute with one another. [A,B]=0 is therefore a prerequisite for any QND experiment.

The cavities in which the atoms are kept must also be high-q cavities that are tuned on or very near resonance with the atom transition. This means that the system is in the strong coupling regime (γ,κ<<g<<ωeg,ωk), and the transitions of the atom are determined by the Jaynes-Cummings Hamiltonian:

$$\hat{H}=\hat{H}_A+\hat{H}_F+\hat{H}_{AF}=\frac{\hbar}{2}\omega_{eg}\hat{\sigma}_{z}+\sum_{\vec{k},\mu}\hbar\omega_k\hat{a}_{\vec{k},\mu}^{\dagger}\hat{a}_{\vec{k},\mu}+\sum_{\vec{k},\mu}\hbar(g_{\vec{k},\mu}\hat{\sigma}_{+}\hat{a}_{\vec{k},\mu}+g_{\vec{k},\mu}^{\star}\hat{a}_{\vec{k},\mu}^{\dagger}\hat{\sigma}_{-})$$

Where $$\hat{H}_A,\hat{H}_F,\hat{H}_{AF}$$ represent the atom, field, and atom-field-interaction hamiltonians, respectively, ωeg is the frequency difference in the atom transition, ωk is the resonant frequency of the cavity, $$\hat{\sigma}_{z}$$ is the Pauli z-matrix, $$\hat{a}_{\vec{k},\mu}^{\dagger},\hat{a}_{\vec{k},\mu} $$ are respectively the creation and annihilation operators for the photon's Hilbert space, and g is the coupling strength.

The atoms that interact with the photons must be optically pumped towards a transition such that they are effectively two-level atoms. The pumping places the atoms initially in the excited state, $$ |\psi(0)\rangle=|\uparrow_{z}\rangle $$, and the subsequent Raman pulse rotates the Bloch vector to $$|\uparrow_x\rangle=\frac{1}{\sqrt{2}}(|\uparrow_z\rangle+|\downarrow_z\rangle)$$, which constitutes the superposition of states in the {z}-basis.

Reflection by a photon in this prepared state activates a rotation of π about the z-axis, changing the state of the atom to $$|\downarrow_x\rangle=\frac{1}{\sqrt{2}}(|\uparrow_z\rangle-|\downarrow_z\rangle) $$ This causes the disparity in response to the final Raman pulse, changing an atom with no or even number of reflections to the ground state $$|\downarrow_z\rangle$$, and an atom with an odd number of photon interactions back to the original excited state $$|\uparrow_z\rangle$$.

Issues

An effective QND experiment is difficult to build as it requires many finely tuned components that are highly sensitive to the surroundings. While this could be incorporated as a feature in fields such as quantum metrology, its application could add more complications than benefits.

A QND detector is also not certain to detect a photon every time. Rempe's maser experiment reported a "click" or detection of a single photon with 74% efficiency, whereas Welte's optical photon QND detectors each succesfully counted a photon between 81%-87% of the time. Additionally, not all clicks can be associated with an actual detection. Detuning in the cavities, imperfect timing with the Raman pulses, and 'premature' atom decay can all contribute to noise, frequently manifested as dark counts. This is best combated by utilization of multiple QND detectors. This increase in redundancy also comes with a drawback: the decreased probability of photon survival in the system.

Despite these hurdles, quantum nondemolition measurements continue to show promise for future applications not only in quantum optics, but also precision metrology, gravitational wave detection, astronomy, and quantum information sciences.

References