When an atom, molecule, or other quantum system is prepared in an excited state, it doesn’t stay there forever. Sooner or later, with no apparent provocation, it decays into a lower-energy state and releases its excess energy, usually in the form of a photon. Although such spontaneous emission may seem like an inherent feature of the material system, it depends just as much on the vacuum into which the photon is emitted. That dependence is quantified by Fermi’s golden rule: The probability of a transition is proportional to the density of final states—including the available photon states of the right energy.
Experiments by Serge Haroche and others in the 1980s showed that one can tinker with the rate of spontaneous emission by placing the emitting system in an optical cavity with a high quality factor. For a transition with energy near a cavity resonance, the density of available photon states—and thus the spontaneous emission rate—is enhanced. For transitions far from any resonance, spontaneous emission is suppressed.
Those experiments, the beginnings of the field of cavity quantum electrodynamics, focused on transitions between electronic energy levels in atoms. (See Physics Today, December 2012, page 16, and the article by Serge Haroche and Daniel Kleppner, Physics Today, January 1989, page 24.) Now, a team of researchers led by Patrice Bertet and Audrey Bienfait (shown in figure 1) from the Atomic Energy Commission in Saclay, France, in collaboration with the groups of John Morton at University College London and Thomas Schenkel at Lawrence Berkeley National Laboratory, have demonstrated the same effect in a different system: the electron spins in a solid.1 Using a superconducting resonant circuit, they’ve increased the spin-flip transition’s spontaneous emission rate by a factor of nearly a trillion.
Figure 1. Patrice Bertet (left) and Audrey Bienfait (right) tune the microwave setup they used to excite electron spins and detect their spontaneous emission.
Figure 1. Patrice Bertet (left) and Audrey Bienfait (right) tune the microwave setup they used to excite electron spins and detect their spontaneous emission.
Cavities and circuits
The new work comes close to realizing the first theoretical description of cavity-enhanced spontaneous emission. In a 1946 conference abstract, NMR pioneer Edward Purcell laid out a proposal for accelerating the relaxation of nuclear spins.2 When the spins are placed in a cavity, he said, the rate of spontaneous emission would be boosted by a factor proportional to the cavity’s quality factor and inversely proportional to its volume.
It was an ambitious suggestion. Even under the strong magnetic field of a modern conventional NMR system, the energy difference between nuclear spin states is tiny—around a millionth of an eV or less—and the density of photon states in free space is proportional to the photon’s energy squared. That density of states, combined with the weakness of the nuclear magnetic moment, yields a spontaneous-emission lifetime longer than the age of the universe. Unsurprisingly, then, spontaneous emission is overwhelmed by nonradiative mechanisms for relaxation, such as coupling to phonons. To turn it into the dominant relaxation process would require a cavity of extraordinarily high quality factor and minuscule volume.
A hint of what was possible came in 1985, when John Clarke and colleagues observed evidence of spontaneous emission in chlorine nuclear spins.3 They placed their sample, a sodium chlorate (NaClO3) powder, not in a closed cavity but inside the coiled inductor of an LC circuit. Like a cavity, the circuit alters the allowed electromagnetic modes in the surrounding space: It enhances the density of states near resonances and depletes it elsewhere. From a small bump in the measured noise spectrum, the researchers concluded that they’d reduced the spontaneous-emission lifetime from trillions of years to about 100 million. But because their sample contained some 1021 spins, enough of them emitted photons over the course of the experiment to produce a visible signal.
Electrons have magnetic moments some three orders of magnitude larger than nuclei. So when subjected to a magnetic field in an electron spin resonance (ESR) experiment (the electron’s equivalent of NMR), they acquire a splitting of tens of µeV, which equates to a frequency of several gigahertz. All told, their spontaneous-emission lifetime in free space is a relatively brief 10 000 years.
Seeking single spins
Bertet and company didn’t set out specifically to enhance spin relaxation, although they always knew it was a possibility. Rather, their primary aim was to design an ESR experiment with single-spin sensitivity. They planned to create a nanometer-scale high-quality LC circuit, small enough to couple to a single electron spin. A circuit’s dimensions function in a similar way to the cavity volume in Purcell’s proposal: The smaller the circuit, the more strongly spins couple to it. And stronger coupling results in both greater sensitivity to tiny signals and greater potential enhancement of spontaneous emission.
As a first step toward the nanoscale, the researchers designed an LC circuit with micron dimensions,4 as shown in figure 2. The capacitors took the form of intermeshed combs, and the inductor was not a coil but a straight wire. When a microwave pulse excites the circuit at its resonant frequency, an oscillating magnetic field encircles the inductor and couples to spins in the immediate vicinity.
Figure 2. A micron-scale aluminum LC circuit, composed of a straight-wire inductor (green) and comb-shaped capacitors, is fabricated on the surface of a bismuth-doped silicon sample. When the circuit is excited at its resonant frequency of 7.3 GHz, the oscillating magnetic field (brown) around the inductor interacts with the nearby Bi–electron spin pairs. The circuit’s presence alters the spectrum of allowed photon states the spins can couple to and thus the spins’ rate of spontaneous emission. (Adapted from ref. 1.)
Figure 2. A micron-scale aluminum LC circuit, composed of a straight-wire inductor (green) and comb-shaped capacitors, is fabricated on the surface of a bismuth-doped silicon sample. When the circuit is excited at its resonant frequency of 7.3 GHz, the oscillating magnetic field (brown) around the inductor interacts with the nearby Bi–electron spin pairs. The circuit’s presence alters the spectrum of allowed photon states the spins can couple to and thus the spins’ rate of spontaneous emission. (Adapted from ref. 1.)
The circuit was made out of aluminum, a low-temperature superconductor that Bertet’s group has considerable experience in working with. But Al has a relatively low critical field of 15 mT; when placed in a magnetic field stronger than that, its superconductivity—crucial for achieving a sufficiently high quality factor—is destroyed. To compensate, the researchers needed to use a spin system with a large energy splitting even at low magnetic field.
Bismuth-doped silicon-28 has the necessary unusual property. At low temperature, each spin-9⁄2 Bi nucleus pairs with a spin-½ electron; the interaction between the two particles creates two available states—one with total spin 4 and one with total spin 5—that differ in energy by 7.4 GHz even with no applied magnetic field. Applying a field of a few millitesla lifts the degeneracy of the Bi nuclear spin to create a manifold of allowed transitions between 7.2 GHz and 7.6 GHz; adjusting the applied field tunes the spin transition energies into and out of resonance with the LC circuit.
Incidentally, nitrogen–vacancy centers in diamond have the same property, with zero-field splitting around 3 GHz. “We could also have done the experiment with NV centers,” says Bertet, “and if we used a metal with a higher critical field for the circuit, it would be straightforward to extend to any other electron-spin system.”
The Si sample was doped with a low level of Bi so the paired spins would be separated widely enough to behave independently; some 104 spins were close enough to the inductor to couple to it. Using a pair of antennas to send and receive microwave pulses at the circuit’s resonant frequency, the researchers excited the spins and subsequently measured their state. From the strength of their signal, they determined that the spin–circuit coupling should be strong enough to make spontaneous emission the dominant mechanism for spin relaxation.
To prove it, they measured the emission rate directly: To a sample cooled to 20 mK, they applied a microwave pulse to excite all the spins, waited an adjustable time T, then measured the fraction that remained in the excited state. In the absence of enhanced spontaneous emission, the excited-state population decays exponentially, via nonradiative mechanisms, with a lifetime of about half an hour. But when the spin-flip transition energy was tuned into resonance with the circuit, the excited-state lifetime dropped to just 0.35 s.
Nor was the enhancement all or nothing. By slightly detuning the transition from the circuit resonance, Bertet and company achieved a range of intermediate excited-state lifetimes, as shown in figure 3.
Figure 3. Tuning the energy of the spin-flip transition to exactly match the circuit’s resonant frequency causes excited-state spins to relax rapidly, as shown by the blue data. When the transition energy is tuned slightly off resonance, the excited-state lifetime increases, as shown by the data in green, orange, and black. (Adapted from ref. 1.)
Figure 3. Tuning the energy of the spin-flip transition to exactly match the circuit’s resonant frequency causes excited-state spins to relax rapidly, as shown by the blue data. When the transition energy is tuned slightly off resonance, the excited-state lifetime increases, as shown by the data in green, orange, and black. (Adapted from ref. 1.)
Reset button
Although specific applications may be a long way off, enhanced spontaneous emission from electron spins may find use in numerous areas. Of most interest to Bertet and colleagues is the ability to make ESR experiments faster and easier. In the current state of the art, once some spins are excited, there’s no easy way to return the system to thermal equilibrium to begin again. Circuit-enhanced spontaneous emission could function as a kind of reset button that can be activated on demand.
A similar benefit could accrue to the field of quantum information. Normally, it’s desirable for qubits to retain their state for as long as possible, but sometimes it’s necessary to wipe the slate clean and start again.
Finally, the results may be applicable to the NMR technique of dynamic nuclear polarization. Because an atom’s nuclear spin states are so close in energy, even under a strong applied magnetic field, it can be difficult to coax more of them into one state than the other. In a typical MRI, for example, nuclear polarization is measured in parts per million. Electron spins are easier to polarize because of their stronger magnetic moments, and under some circumstances it’s possible to transfer electron polarization into nuclear polarization via a so-called flip-flop transition. The Bi–electron spin pairs undergo just such a transition—due to selection rules, flipping the electron spin necessarily changes the direction of the coupled Bi nuclear spin. It’s possible that other flip-flop transitions could also be accelerated on demand.
Bertet and colleagues’ next step is to follow their original plan and work toward a nanoscale resonant circuit. The smaller dimensions promise not only single-spin ESR sensitivity but even faster spontaneous emission.
