Locking “without” light

Relative frequency drift As explained earlier, we have the possibility to stabilise the science cavity by locking only the auxiliary. Although the fix connection works fine at high frequencies, low frequency relative drift in cavity lengths is inevitable.

One can simply imagine that if the two cavities are not perfectly parallel, a given piezo displacement results in a different change in cavity lengths. This is naturally correlated with the thermal deformation of the bridge, as the piezo displaces accordingly to keep the cavity length locked. In a similar measurement, we record both cavity resonances while scanning the piezo in a experimental cycle. The relative drift between the two cavities also follows the thermal cycle of a single cavity (Fig.2.30(b)). Though seemingly reproducible, a simple feed-forward correction of this relative drift is not sufficient. In fact, there is a slow random drift over ∼ 10 MHz in a few seconds, on top of the more deterministic thermal cycle. We do need to correct the cavity length locked through the auxiliary cavity. This is achieved through an additional frequency lock.

Two-cavity “interlock” Let us first recall how the cavities are locked. The PDH signal of the science cavity locks the cavity length to the 1560 nm laser, which is referenced to an absolute frequency. At this particular cavity length (of the science cavity), the auxiliary cavity is detuned 35 GHz from the 1560 nm laser. We use a third-order sideband to bridge this gap.

The relative drift between the two cavities can be corrected via this 35 GHz side-band. More concretely, we can lock both PDH signals of the two cavities simultaneously (Fig. 2.31(a)): the piezo is controlled by the PDH lock of the auxiliary cavity, whose reso-nance frequency (via the 35 GHz sideband) is controlled by the PDH signal of the science cavity, such that the latter has the correct frequency.

In practice, a precise tuning of about 40 MHz is required on the 35 GHz sideband. While a direct frequency modulation (FM) is not available, we instead reference the synthesiser externally (10 MHz) with a signal from a DDS that can perform analog FM. The PDH error signal of the science cavity then drives this analog FM after a digital PI controller (Fig. 2.31(b)).

With this two-cavity interlock, the PDH signal of the science cavity serves as a “frequency lock” of the auxiliary cavity. We can keep the auxiliary cavity always locked, but turn off the locking light in the science cavity while keeping the frequency lock at the last locked value (via a sample-and-hold circuit, an example is shown in Fig.2.32). This is a practical scheme for locking the science cavity “without light”.

Additional feedback Once the frequency lock is turned off, the science cavity is again subject to the relative drift. But at least we start from the right place. In practice, we would be interested in turning off the locking light only during the critical Ramsey time, to minimise the light shift from the locking light. In fact, with the two-cavity interlock, the relative drift can be precisely assessed by opening the frequency lock but keep tracking of the PDH signal of the science cavity. We observe that during the Ramsey sequence of the experimental cycle, the relative drift is a few MHz in 100 ms.

Nevertheless, we realise that the relative drift is also correlated with the correction applied to the piezo (by the PDH lock of the auxiliary cavity). This can be understood again in the picture with non-parallel cavities: the relative length change is directly related to the piezo displacement. The correction signal to the piezo therefore provides additional information about the drift. We add the piezo correction signal (PDH of the auxiliary cavity) to the correction signal for the frequency lock (PDH of the science cavity) with a properly tuned

PZT

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Figure 2.31 Two-cavity interlock setup. (a) shows the simplified schematics of the two-cavity interlock. (b) shows the detailed setup (only the lock part). Electronics are shown in grey, while the PDH signals are in the same colours as in (a). Most of the signal processing is performed by the Red Pitaya board. The additional feedback explained in the text is indicated in orange.

gain (block diagram shown in Fig. 2.31(b)). Note that it plays no role when the two-cavity interlock is on, but starts to correct the relative drift after the frequency lock is open (Fig. 2.32).

The final residual relative drift is in the order of 1 MHz in 100 ms, and appears rather reproducible. In principle this could be further improved by additional feed-forward or calibration.

0.00 0.02 0.04 0.06 0.08 Time (s)

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frequency lock

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Figure 2.32 Demonstration of additional feedback of the piezo correction signal to the frequency lock. It is actually a sequence used to load atoms in the intra-cavity lattice (Sec.3.2.4). Here the cavities are locked with the auxiliary cavity and we plot the trans-mission and the PDH error signal of the science cavity, as well as the piezo control signal – output of the PDH lock of the auxiliary cavity (HF). The two-cavity interlock is active in the orange region for 10 ms, then the frequency lock holds the last value (left panel). After the grey region (during which the 1560 nm power in the science cavity is being ramped up), we see the relative drift from the PDH error signal (red) and the transmission (black).

Clearly, the piezo control signal is correlated with the residual drift, meaning that it could be used to correct the drift. With this additional signal applied to the frequency lock (right panel), the residual drift is clearly reduced. While at this timecale (50 ms) the correction seems perfect, we can see residual drift at longer timescale of hundreds of ms. The final kink in the signals is due to a perturbation in the sequence.