Jonathan Miners Kindem

Thanks to the Faraon group: Tian for bringing me up to speed in the lab and teaching me so much over the years. The whole flat optics subgroup for listening to all the rare-earths talk over the years and for your friendship. Thanks to friends from Caltech who helped me escape from the lab on occasion: The Steuben House for all the parties.

Thanks to all the wonderful administrators at Caltech who keep things running so smoothly.

INTRODUCTION

This is primarily a review of the Jaynes-Cummings model in the bad-cavity limit and Purcell enhancement. The interesting and unique properties of the rare earth ions are due to the localized nature of these 4f orbitals. In the case of rare earth species, the magnitude of the Coulomb and spin-orbit coupling are of the same magnitude.

These devices were initially designed and manufactured by one of the former postdocs in the lab, Dr.

Figure 1.1: Example of the photonic crystal nanobeam resonators used in the Faraon group

NANOPHOTONIC INTERFACES IN ND:YSO AND ND:YVO

We showed that this e↵ect can be used to store and retrieve frequency qubits in collective ensemble excitation with high bandwidth (⇠ 50 GHz) and high retrieval reliability (98.7%). The first step towards this goal was to confirm that optical coherence was still preserved in Nd:YVO devices. This work was followed by a more complete investigation of the AC Stark shift in the ensemble, the results of which are published in [55].

In the same device used for work with quantum memory, we demonstrated the optical detection of individual Nd ions in YVO.

Figure 2.1: Highlights from [1]. a) Lifetime measurement of cavity-coupled ions with cavity on resonance (red) and detuned (blue) showing lifetime reduction from 254 µs to 87 µs

SPECTROSCOPY OF 171 YB:YVO 4

In this work, we focus on the coherence properties of optical and nuclear spin transitions in the regime where the linear Zeeman interaction is dominant. The inhomogeneous linewidth of the nuclear spin transition was measured using continuous wave Raman heterodyne detection [83]. The inhomogeneous broadening of the nuclear spin transition was measured using continuous wave heterodyne Raman spectroscopy [83].

The symmetry of the crystal gives rise to the strong selection rules on the optical transitions.

Figure 3.1: Energy level diagram for 171 Yb:YVO 4 . a) Crystal field splittings of

THEORY: AN ION IN A CAVITY

We will assume that the atom is located in the maximum of the electric field of the cavity with an optimal alignment between the polarization of the cavity and the dipole. In the case of no input field (ˆain = 0), the cavity field is given by Eq. 4.6) the field at the exit from the cavity is given by ˆ. From the above we see that the decay rate of the atom in the cavity has been changed from the free space value to 2.

The ratio between the decay rate of the emitter in the cavity and the decay rate of the emitter in free space is In the literature, both Fp and Fp are often referred to as the Purcell factor of the system. We see that the presence of an ion in the cavity changes the reflection spectrum for the cavity.

Experimentally, we are interested in extracting the cooperativity (⌘d) by measuring the change in the power reflection spectrum of the cavity due to the presence of the ion. A gold coplanar waveguide was fabricated next to the device to enable microwave manipulation of the ions in the optical cavity. This is then combined with the output of the low power path of the Ti:Sapph and sent to the device.

The other laser path is sent to the o↵ array closure configuration described below. This slow circuit adjusts for any displacement in the Ti:Sapph relative to the reference cavity and allows scanning of the laser when locked. First, we have the ion emission fraction in the cavity mode, pcav.

A simplified schematic of the microwave setup for the current measurements in the Yb-171 singles devices is shown in Fig.

Figure 4.1: Jaynes-Cummings model of two-level emitter in a cavity. A two-level emitter with transition frequency ! a and free space decay rate is coupled with single photon Rabi frequency 2 g to a cavity with frequency ! c

MEASUREMENTS OF A SINGLE YB ION IN YVO

1 Another option to explore in the future would be to measure the change in the cavity transmission or reflection due to the presence of the ion [10]. In this case, the measured count rate can be mapped directly to the coupling strength of the ion to the cavity (See Chapter 4). Based on this lifetime, the fraction of the ion emission is in the cavity mode.

In our system, the lifetime of the ion in the cavity (4 µs) is much longer than the last detection setup time (50 ns). As a check that this is a reasonable assumption, we can calculate the probability of two photons arriving within the last detector time given the lifetime of the ion in the cavity. We can look at the behavior of the optical transition in an applied magnetic field to further confirm the isotope of the ion.

If we assume instead that the spin relaxation is negligible, we can get an estimate of the optical branching ratio. A more direct measurement of the optical branching ratio will be performed in the next chapter. After verifying that we are moving the population between two ground states of the ion in the cross-correlation measurements, we want to initialize the population to a single ground state.

Further investigations are needed to investigate the magnetic field dependence of the optical coherence times in this ion. Note that the frequency refers to the cavity-locking frequency and not to the denomination from the center of the ion.

Figure 6.1: Integration time required to achieve a SNR of 20 as a function of de- de-tected count rate due to ion emission for increasing background count rates (in counts per second).

MEASUREMENTS OF SINGLE 171 YB:YVO 4 IONS

In the high-field limit, however, the optical transitions of isotope 171 overlap with those of isotope 173. Changes in the design of the optical and microwave cavities used should allow further exploration of this regime in the future. I will call these two states the qubit subspace in the rest of the text.

We will use the lowest frequency optical transition (A) to read the state of the ion. Depending on the resulting site symmetry of the ion, there is a chance that the ion now exhibits a strong DC shift [127]. Now that the ion has been initialized to state |1ig, we turn to measurements of its optical coherence properties.

The ability to perform coherent control pulses on the optical junction also allows for further measurements of the optical coherence lifetime. A complementary measurement to do in the future would be to measure the optical coherence time of the ion as the cavity is detuned from resonance. We are currently implementing a modification to the SSRO procedure to increase the readout fidelity of the ion in|0i.

In addition, this method checks whether the ion was actually in the subspace of interest at the beginning of the measurement. If the ion starts in |auxistat, we will get a zero count in both rounds of reading.

Figure 7.1: Zero-field energy level structure of 171 Yb:YVO 4 reproduced from Chapter 3 for reference

FUTURE DIRECTIONS

600 mK and a magnetic field of 440 mT along the c-axis of the crystal in the sample with a significantly higher doping density (100 ppm). As mentioned above, one of the next major steps towards this goal will be the demonstration of indistinguishable photon emission. After showing improvements in coherence lifetimes, we want to show a direct measure of indistinguishability.

To measure the indistinguishability between a pair of photons from a single emitter, we can introduce an additional delay line in order to interfere photons from successive excitations of the same emitter. For this, we need the delay line to be significantly longer than the ion lifetime. With the additional insertion loss of the necessary beam splitters and fiber links (⇡ 1dB total), this would bring the total detection efficiency from 1% to ⇡ 0.3% for a two-photon detection probability of 9 ⇥ 10 6 .

Of course, such measurements cannot be performed until the remaining questions about the properties of the system are resolved. We can understand the degree of entanglement dispersion we can achieve within the laboratory environment with reasonable improvements in equipment. This then corresponds to a success rate of ⇠31 Hz for two ions in the same lab.

Further investigation of the ultimate limits of performance and stacking efficiency in these devices is warranted before making any further speculations. We then used the zero-field level structure of 171Yb:YVO4 to demonstrate high-fidelity one-shot readout of the spin state.

BIBLIOGRAPHY

Becher, “All-optical control of the silicon-vacancy spin in diamond at Millikelvin temperatures,” Physical Review Letters. Afzelius, “Coherent spin control at the quantum level in an ensemble-based optical memory,” Physical Review Letters. Afzelius, “Towards highly multimode optical quantum memory for quantum repeaters,” Physical Review A - Atomic, Molecular and Optical Physics.

Faraon, “Control of rare earth ions in a nanophotonic resonator using ac Stark shift,” Physical Review A97, 1–. Sridharan, “Cavity QED treatment of interactions between a metal nanoparticle and a dipole emitter,” Physical Review A - Atomic, Molecular and Optical Physics. Lagendijk, “Resonant scattering and spontaneous emission in dielectrics: Microscopic derivation of local field effects,” Physical Review Letters.

Tapster, “Intensity fluctuation spectroscopy of small numbers of dye molecules in a microcavity,” Physical Review A - Atomic, Molecular and Optical Physics. Prawer, "Photophysics of chromium-related diamond single-photon emitters," Physical Review A - Atomic, Molecular, and Optical Physics. Hanson, "Control and Coherence of the Optical Transition of Single Nitrogen Vacancy Centers in Diamond," Physical Review Letters.

Reducing Decoherence in Optical and Spin Transitions in Rare Earth Doped Materials”, Physical Review A. Hanson, “Detection and Control of Individual Nuclear Spins Using a Weakly Coupled Electron Spin”, Physical Review Letters.

TRANSITION SELECTION RULES IN 171 YB:YVO 4

We then want to write the spin-orbit multiples (2F5 .. 2) in terms of irreversible representations of this symmetry. In our system, we find that the 984 nm optical transition exists for both ⇡ and ⇡ polarizations, which implies that one state is 6 and the other is 7. That is, by introducing nuclear spin we split into three levels in the ground state and the excited as noted.

We can then derive the electric dipole transition selection rules for zero field states using e.g. From this, we see that at zero field we expect three⇡-polarized transitions and four-polarized transitions, which is what we observe. This then helps assign the order of energy levels based on the observed transitions.

We observe that the highest energy optical transition splits into four in an applied magnetic field, meaning that the degenerate state|1±1i is lowest in energy in the ground state and highest in energy in the excited state. This ambiguity can be resolved from the measured values of g and A, which (through the well-considered corollary of Eliot and Stevens) establish the order of these states. In terms of our spin Hamiltonian, we can write the electric dipole operator for light-polarized longitudinal waves.

From above we see that ˆPz transforms as 4 and we can do the same for the other polarizations. The contribution of the nuclear spin to the optical transition strength is then given by the overlap of these states.