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Superradiance for Optical Clock Transition

Summary:

Optical clock is an instrument that can precisely measure time differences using an ultra-stable atomic transition as a frequency reference. Accuracies much better than parts per billions are typical. However, due to frequency instability in the lasers used to probe the atomic transition, today’s best optical clocks can only resolve a much broader linewidth than the fundamental linewidth of the clock transition. Therefore, the quality factor (Q) is spoiled due to the limitations imposed by the probe laser.

To approach the fundamental linewidth of the atomic transition, the atomic transition itself feeds to a laser cavity, ignoring the other possible modes allowed by the laser cavity. The application of generating a superradiant laser pulse is shown to realize this possibility.

Superradiance:

When a collection of atoms collectively interacts with an external field, the rate of photons coherently spontaneously emitted are proportional to the N^2 atoms. The separation of the atoms must be smaller than the radiation wavelength, yet farther than the particle wavelength away. This condition allows the radiotin of one atom to affect another, while also eliminating dipole-dipole interactions between atoms. Another key requirement is that the emission rate must be larger than the decoherence rate.

In the case of incoherent spontaneous emission, since the phases radiated by N atoms have no relation to each other, the light intensity of spontaneous emission will only be proportional to the atomic number N of excited states and provides a continuous output of photons. Achieving a rate proportional to N2 atoms is the key distinction of superradiation.

In 1954, Dicke developed the theoretical prediction of superradiance of neutrons inside a uniform magnetic field [1]. Instead of the atoms not interacting with each other, now they are close enough to have their interaction with the incident field be coupled, whilst the atoms having no other interactions with each other. Dicke’s theory offered the opportunity of using superradiance in optical clocks.

Optical clock elements have quantum states with extremely long lifetimes, such as 87Sr, which has roughly 150 s for its lifetime. The characteristic of long lifetime for optical clock elements corresponds to a very narrow linewidth of 1 mHz, which is orders of magnitude narrower than a typical laser. However, the optical clock linewidth is limited by the probe laser frequency instability, which is caused by thermal Brownian motion. Thus, the quality factor of the optical clock is strongly influenced by probe laser.

A method to remedy the losses due to the probe laser and closer approach the atomic transition linewidth is to directly collect the light from a long-lived quantum state. Thompson shows us how it is done [2]. Noise from the probe cavity that would normally broaden the laser linewidth are ignored by the atomic transition driving the laser mode. This is a rather difficult task to do with a very weak atomic transition. Using superradiance, a significant improvement in the lifetime now reaching 100 ms vs the 150 s. This makes the practice of using the atomic transition as the driving force of the optical clock feasible.

Collective Bloch Sphere:

The group of atoms are treated as a single quantum system. The dynamics of the atomic transition can be described using a collective Bloch sphere. With all the atoms excited the Bloch vector points up at |e>. When electrons are in the atom’s ground state, the Bloch vector points down. The magnitude of the projection of the Bloch vector on the equatorial plane is proportional to the rate at which atoms are decaying into the quantum cavity. The impact of this process is that the electric field radiated is proportional to the N atoms, and thus the power is proportional to N2 atoms. This ‘super’ radiation is similar to a pulsed vs CW laser, in which the action of the pulse only happens for a sort of duration, but with much higher intensity.

Thompson’s Experiment:

To overcome the current research challenge of reaching the atomic linewidth, Thompson and his research group proposed an approach to achieve pulsed superradiant lasing for the first time on an ultraweak optical clock transition.

Firstly, Thompson’s team prepared the experimental system by setting atoms in |e〉. They optically pumped the atoms to |g〉, the F = 9/2, mf = 9/2 sublevel of 1S0, and then adiabatically transferred up to 75% of the atoms to |e〉 using a frequency-swept 698-nm transfer beam applied through the cavity. In addition, they briefly applied lasers to the dipole allowed 1S0-to-1P1 transition to heat any atoms remaining in the ground state out of the lattice to prepare the atoms with full inversion and ensure the trigger of superradiance pulses.

Secondly, Thompson’s team varied the number of atoms up to N = 2.5 x 105 and measured the photon output rate and pulse width during the experiment. Figure below show the data collected of the superradiance pulses for varies atom numbers. In the figure, the narrowing of the pulse with atom number provides a measure of the enhanced decay rate. The more atoms participating the superradiant pulse rapidly expedite the decay process. This appears like positive feedback system, where the more atoms there are, the more the atoms talk to each other and are convinced to decay together. As noted in the pulse peak shifting to a shorter time, the atoms agree to start the decay much sooner.

Due to decoherence of excited atoms, there is a threshold for number of atoms needed (Nt) to achieve the avalanche effect of stimulating the superradiant pulse. The peak photon output rate vs number of atom is given as:

Where  is the number of photons in the lattice in excess of the threshold number;

Where  is the probability of an emitted photon coupling to the cavity;

Where  is set by the requirement that the collectively enhanced decay rate exceeds the atomic decoherence rate.

Decay rate scales quadratic with  as: Factor of an atom to leave cavity (C = 0.41).

Thirdly, the output of their experiment shows an enhancement of the decay rate from 150 s to under 100 ms. What’s more, the emitted laser light both serves as an absolute frequency reference and offers a new path toward lasers with linewidths at or below the millihertz level, orders of magnitude narrower than what has previously been achieved with traditional optical reference cavities.

This experiment successfully demonstrated an ultranarrow optical transition can be made to lase in a pulsed manner, with each atom emitting up to a single photon. The promising of this work provides many advantages replenishing atoms lost to heating or collisions.in addition, Thompson’s study shows that superradiant laser is its reduced sensitivity to fluctuations in the length of the laser cavity. The dominant contribution led to stimulated emission in a regime where the cavity field is much shorter-lived than the coherence of the atomic ensemble, and open new avenues for the improvement of optical clocks, ultra-stable lasers, and other atomic sensors along with their many applications.

References:

1.    Dicke, R. H. (1954). Coherence in spontaneous radiation processes. Physical review, 93(1), 99.

2.    Norcia, M. A., Winchester, M. N., Cline, J. R., & Thompson, J. K. (2016). Superradiance on the millihertz linewidth strontium clock transition. Science advances, 2(10), e1601231.