CAPITULO IV De la Época de la Quiebra
TITULO PRIMERO Disposiciones Generales
Chapter 2 introduced two important relaxation processes in rare earth systems: spin-lattice and spin-spin relaxation. The first process, namely spin-lattice relaxation, was character-
ised by the measurements in the previous two sections. The second process was masked in the previous measurement because it did not redistribute the hyperfine population, and therefore did not change the population envelope. However, it will affect any measure- ments operating on a subgroup of the entire ensemble. This is particularly the case for spectral holeburning, which is a vital component of many quantum memory applications. Therefore, it is important to characterise the cross-relaxation process.
This section presents a study of cross-relaxation, through measurements of spectral hole lifetimes as a function of magnetic field along the D1 optical extinction axis. For these measurements the 167Er ensemble was initialised in thermal equilibrium at 1.4 K, instead of a hyper-polarised state. Spectral features were then burnt into the centre of the
∆mI = +1 absorption band, in order to avoid isotopic impurities while maintaining good
optical depth. It should be noted that all three bands overlapped (to varying degrees) for fields less than 1T, as indicated by Figure 2.3.
-2 0 2 Frequency (MHz) -10 -8 -6 -4 -2 0 Modul ation resp onse (dB ) 1.5 secs 4.5 secs 7.5 secs 10.5 secs 13.5 secs 1 secs 11 secs 21 secs 31 secs
0.1 T
-10 -5 0 5 10 -6 -5 -4 -3 -2 -1 02.5 T
Figure 4.17: PM spectra of features burnt at 0.1T and 2.5T. The modulation response is inverted with respect to AM spectroscopy (see Figure 3.4). While the trench at 2.5T is wider, it’s sharp edges illustrate reduced spectral diffusion. Some ringing is also evident at the sharp edges of the trench, caused by the fast sweep of the tracking generator.
For fields greater than 1.5T, 10 MHz wide spectral trenches were burnt by sweeping the EOM sideband over a 10 MHz range. This width was chosen as it was considerably broader than the laser linewidth of 100 kHz, resulting in a flat trench. This sweeping was performed for 10 seconds to reach steady state spin-redistribution, as the duty cycle of the sweep was only 10 Hz. A fixed time after a trench was burnt, PM spectra were recorded
with a 50 ms sweep of the EOM sideband over the trench. This burn-scan process was repeated for several fixed delays (up to 6) to determine the trench lifetime, and the series of trenches burnt at 2.5T are shown on the right hand side of Figure 4.17 as an example.
Faster relaxation and increased hole broadening made it difficult to burn deep trenches for fields less than 1.5T. At these low fields spectral holes were instead generated using a fixed-frequency RF source. Even with a fixed-frequency EOM sideband, however, the holes broadened to several MHz due to spectral diffusion (see Section 3.1.2). As an example, the holes burnt at 0.1T are shown on the left hand side of Figure 4.17. For both the trench and hole burning, the laser carrier was 1.5 GHz blue detuned from the∆mI= +1band to
prevent unwanted absorption. In both cases lifetimes were determined by fitting a single exponential to the depth of the trench or hole.
0 1 2 3 4 5 6
Magnetic field (T)
10 20 30 40 50 60 70 80Spectra
l Hole Lifetim
e (Seco
nds)
Hole lifetime Theoretical phonon density (Arb. units)Figure 4.18: Blue: The lifetime of spectral features burnt into the∆mI = +1absorption band,
as a function of magnetic field along the D1 optical extinction axis. y-axis error bars are the standard error in the fit to each point. Black dashed: The theoretical phonon density at the electronic transition frequency, defined by Equation (2.12) and equivalent to the Orbach process in Equation (4.3).
The trend in lifetimes shown in Figure 4.18 can be explained by three effects domin- ating at different magnetic fields. In order of increasing field, these are electronic cross- relaxation, electronic spin-lattice relaxation and hyperfine cross-relaxation. The following three subsections outline the justifications for these conclusions.
4.10.1 Electronic cross-relaxation
Rapid electronic cross-relaxation between neighbouring Er ions limits the electronic spin lifetime at low field in Er:YSO [98]. This also shortens hyperfine transition lifetimes to100
ms in low field, because of the Orbach process described in Section 2.4.4 [69].
Fortunately, electronic cross-relaxation can be suppressed by increasing the magnetic field. In an applied field the Zeeman splitting of the Kramers doublet causes the electron spin ensemble to polarise into theS˜=|−1/2istate, at a rate determined by the Boltzmann factor. This suppresses the electron-spin cross-relaxation because the density of resonant ions in the S˜ = |+1/2i state is reduced. The rate of hyperfine transitions is then also reduced, which causes the increase in hole lifetimes observed between 0 and 0.1T
4.10.2 Spin-lattice relaxation
Beyond 0.1T the spectral holes lifetimes begin to shorten, even though electronic cross- relaxation should be quenched (e−∆E/kT = 0.0006 at 1T and 1.4 K). This suggests that
another process mediates hyperfine transitions beyond this point. In Section 4.9 it was determined that the Orbach spin-lattice process drives hyperfine transitions in large fields at higher temperatures. Because this process is mediated by thermal phonons, it also becomes strong at low temperatures if the magnetic field is reduced. This is illustrated by the black dashed trace in Figure 4.18, which plots the density of thermal phonons at the electron spin transition frequency (Eq 2.12). Moreover, the trend in lifetimes between 0.1- 3.0T is anti-correlated with this distribution. Based on this observation and the results of the previous section, it was inferred that Orbach spin-lattice relaxation dominates between 0.1 - 3.0T
4.10.3 Hyperfine cross-relaxation
Figure 4.15 from the previous section shows that direct nuclear spin-lattice relaxation limits the hyperfine lifetime at high fields, resulting in a hyperfine T1 of 12 minutes at 1.4 K. However, this lifetime is an order of magnitude longer than the ∼60 sec plateau in
hole lifetimes observed here. This implies that spin-lattice relaxation no longer limits hole lifetimes above 3T. Instead, we infer that hyperfine cross-relaxation is dominant based on several observations.
Firstly, the complicated field dependence above 3T suggests some form of resonant coupling. For hyperfine cross-relaxation, this is consistent with a change in the density of resonant nuclear spins as a function of magnetic field. For certain magnetic fields the hyperfine transition energies of site 1 will be sufficiently similar to those of site 2, such that hyperfine cross-relaxation between the sites can be mediated by Y spins. This increases the effective spin density, causing some regions above 3T to exhibit shorter lifetimes.
Furthermore, this is the only remaining interaction (of the usually considered mechan- isms described in Section 2.4) which can reduce hole lifetimes below the hyperfine popula- tion lifetime; Electronic cross-relaxation was ruled out due to Boltzmann statistics.
It should also be noted that the lifetimes in Figure 4.18 were measured with the hy- perfine population in thermal equilibrium. Measurements with a hyper-polarised ensemble should yield significantly shorter lifetimes at high field, due to the four-fold increase in resonant spins.
4.10.4 Spectral diffusion
An additional consideration for shaping narrow absorption features is spectral diffusion. Figure 4.17 presents spectral features burnt at 0.1T and 2.5T. While they both exhibit the long lifetimes necessary for efficient holeburning, the spectrum at 0.1T shows increased spectral broadening. This broadening is consistent with the process of spectral diffusion, which was introduced in Section 3.1.2.
More specifically, the increase is due to the change in electron spin dynamics. For fields greater than 2T the electron-spins are highly polarised. In this regime a phonon-mediated
spin flip (spin-lattice coupling) is rapid and short lived; the large density of phonon modes implies that an electron spin will fall back quickly to the ground state. Thus, each electron spin flip contributes only a small contribution to optical transition dephasing.
In a field of 0.1T, however, the electron spins spend appreciable time in either spin state. This produces large dephasing, even though the lifetime (and hence rate of electron spin transitions8) is similar to the 2.5T regime. For this reason quantum memory schemes
8
In the previous subsections it was described how spectral hole lifetimes below 3T are limited by the electronic spin transition rate
requiring narrow linewidths cannot be realised at low fields. Large dephasing will also shorten optical and hyperfine coherence times in the low field regime.