Dynamic decoherence control

The two pulse spin echo technique described above completely removes the decoherence due to static broadening in an ensemble, but it does not remove the decoherence due to dynamic broadening. Dynamic decoherence control (DDC) is a strategy for fighting decoherence due to dynamic broadening. It works by applying a sequence of control fields periodically, usually fast and strong pulses in cycles, to alter the dynamics of the system and refocus the dephased ensemble Bloch vectors. In contrast to quantum error correlation (QEC), for example, DDC does not require additional measurement or memory resources. Hence it can be integrated with other error-avoiding or error-correcting techniques in a straightforward manner to help achieve fault-tolerant control.

The DDC technique works effectively only if the reconfiguration time τ₁ (= 1/R) of the dephasing bath is longer than the time scale τ_c at which pulses can be applied to the system. Then the frequency detunings due to the dephasing environment remain effectively static during each pulse period and are therefore rephased. If the decoherence contribution is rapidly fluctuating, with a correlation time much shorter than the time required for the application of refocusing pulses, then this type of noise cannot be refocused by a DDC pulse sequence.

The discussion above shows that, in order to fight decoherence, DDC repetition rate needs to be faster than the reconfiguration rateR of the dephasing bath. However, there are practical challenges if the DDC pulse rate is too fast. First, in order to make the pulses hard pulses such that the evolution during the pulses is neglected, the DDC pulse rate is required to be lower than the Rabi frequency while the Rabi frequency is often limited by the possible RF power supply. Secondly, a high pulse rate means that the pulse number is huge. Since the precision of any experimentally accessible operation is finite, the number of pulses that can be applied without degrade the signal is limited by the pulse errors. For these two reasons, to realise the DDC effective, it is necessary to keep the pulse rate low, which is only possible when the reconfiguration rate of the dephasing bath is low .

For the system discussed in this thesis, hyperfine transitions in Eu3+:Y2SiO5, the

decoherence is dominated by the magnetic field perturbation due to the random spin flips of the host Y3+ions. Though the Y3+spins flip at a very fast rate at zero field, this flipping rate can be largely reduced in the presence of an external field principally because the field induces a frozen core as discussed in Section 2.5.3. The frozen-core effect divides the host Y3+ ions into two groups, that are bulk Y3+ ions, far away from Eu3+ ions, producing small amplitude but rapid magnetic field perturbations, and frozen core Y3+ ions, near the Eu3+ _{ions, generating large but slow magnetic field perturbations. If, in addition,}

the applied field is chosen such that the transition of interest is a ZEFOZ transition, the rapid and small magnetic field perturbations originating from the bulk Y3+ ions have a negligible effect on the hyperfine transition frequency due to the small field sensitivity. The remaining decoherence contribution is the large field perturbations from the frozen core Y3+ ions. Since the frozen core reconfigures at a suppressed rate, a DDC sequence with relatively low repetition rates should be quite effective in removing the decoherence. To summarize, ZEFOZ removes the small and fast decoherence effect of the bulk Y3+ and

DDC removes the large and slow decoherence effect of the frozen-core Y3+. As we can consider ZEFOZ technique as a high frequency decoherence filter and DD technique a low frequency decoherence filter, their combination is likely to be very effective.

Applying DDC pulses at a ZEFOZ transition have another benefit. If the field is detuned away from a ZEFOZ field, driving a Eu3+ _{transiton using a DDC pulse will}

slightly change the magnetic moment of Eu3+. As discussed in Section 2.5.4, a change in the europium magnetic moment would accordingly cause a reconfiguration of the Y3+bath, which will have a back action on Eu3+. When a large number of DDC pulses are applied, the accumulated error due to this issue can decrease the DD effectiveness. Whereas, if the DDC pulses are applied at a ZEFOZ transition of Eu3+, as discussed in Section2.5.4, driving the Eu3+ transition does not change its magnetic moment, thus will not disturb the Y3+ system. This means that, the combination of ZEFOZ, the frozen core, and DDC can result in significant extension of the coherence times in this system.

Two different DDC pulse sequences were used in the experiments presented in this thesis. The first sequence implemented was the CPMG(Carr-Purcell-Meiboom-Gill) [101] spin echo sequence, which is commonly used in NMR experiments. It provide a simple and effective way to test the DDC effectiveness. As shown in Figure 3.4, the pulse sequence has the following format:

(π/2)x−τc/2−πy−τc−πy−τc−πy−...πy−τc/2−echo.

π/2 π Echo τ 2 τc 2 τc ωt ωt 90° 90° 2 τc 2 τc x N 0° 30° τc τc τc 0° 30° 2 τc 2 τc 30° 2 τc τc ωt KDDx CPMG 90° 0° 0° x N Echo π π/2 π π π π π π τ

Figure 3.4: CPMG and KDDxpulse sequences. Here ωtis the frequency of the driving field,N is the repetition number of the indicated pulse sequence,τc is the pulse interval andτis the total evolution time.

Although the CPMG sequence is commonly used to extend coherence, the method is only effective in preserving quantum states that possess a particular phase. Hence, it is not suitable for use in a practical quantum memory, which is required to store arbitrary quantum states. To test the suitability of the Eu3+:Y2SiO5 system for arbitrary

quantum state storage, another DDC sequence, known as KDDx, was also used. The KDDx sequence is effective in preserving arbitrary states and is robust against errors due to pulse area and off-resonant excitation [102]. It has the general format:

(π/2)π/2+φ−τc/2−ππ/6+φ−τc−πφ−τc−ππ/2+φ−τc−πφ−ππ/6+φ...−ππ/6+φ− τc/2−echo,