B.1 Background
The cavity probe introduces inhomogeneous light shift on the atoms, causing dephasing.
We now perform composite measurements in which two probes are separated by a π pulse, forming an echo sequence. Probing at a fixed detuning κ/2, the transmitted photon numbers in the two probes differ if ∆N ≡ N↑− N↓ 6= 0. This causes a residual dephasing correlated to ∆N (Sz), initiating the amplification effect. Nevertheless, this correlation can be removed by toggling the probe at ±κ/2 respectively for the two probes.
To achieve a good π pulse, we apply composite pulses, whose constituent pulses have to respect the thermal oscillations in ~z, each having a minimum duration of the trap period
∼ 9 ms. The entire ˜π pulse takes 27 ms, and each composite measurement takes about 45 ms. This is clearly a disadvantage. Note that the long ˜π is not necessarily affected by the ISRE. Because at high probe power, the spins are completely dephased after the first probe, ISRE then would not happen. It is only after the rephasing by the second pulse that ISRE starts to play a role.
An alternative solution is to use another cavity field (another longitudinal mode of the cavity, e.g. 1 FSR away) to compensate the probe. this field should produce the same but negative light shift on the clock transition. Having the same intensity profile as the probe (a reasonable assumption), it compensates the probe light shift exactly for all atoms. The experiment can be further simplified if we separate the two cavity fields in time, namely a normal probe pulse followed by a compensation pulse. However, if they are separated in time, they do have to respect the oscillation period such that the transverse oscillations are averaged. The minimum total duration will be 9 ms.
In the following, we first show that the compensation field cannot compensate the light shift exactly in the presence of QPN (in Sz), namely the correlation between dephasing and Sz persists. Possible resolution is discussed. We also evaluate the compensation in the presence of atom number fluctuation and show that this scheme would work in realistic situations.
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Compensation setting
Recall that for the cavity shift by the atoms δωc= g2
ωat/2N↑+ g2
−ωat/2N↓ = 2g2
ωat∆N (B.1)
where we assume equal coupling g of the two clock states, and an effective equal detuning
∼ ωat/2. For the second cavity field, one FSR away (ωfsr= 2πνfsr):
with N the total atom number. Note that g is geometrical, which we assume the same for the two cavity modes.
Simultaneously, the light shifts on the clock transition for both fields read:
δωat = 4g2 where ncand n0care the average intra-cavity photon numbers for the two fields, respectively.
The light shift compensation requires δωat0 = −δωat, giving n0c= 4
G ≈4ωfsr2 /ωat2 ≈1300 is the “gain” needed for the compensation condition. We find that the frequency shift of the compensation mode has been attenuated by the same factor compared to the probe:
We notice that δωc0 also depends on N, susceptible to technical fluctuations.
B.2 Residual dephasing
More importantly, we shall look at the correlation between the light shift and ∆N. It is better to note the phase shift induced on an atom by nc transmitted photons, namely the light shift integrated over the cavity lifetime 1/κ:
φac= δωat κ ≈ 4g2
ωat nc
κ (B.6)
ncis a function of the probe input photon number np(a constant) and the cavity shift. With the probe detuned at κ/2 from resonance, we have
nc= T np≈
We shall assume that the compensation condition is satisfied in the average case, ∆N = 0
For the second mode, we have similarly:
δφ0ac= ∂(δωat0 ) δφ0ac∼ −δφac, we should fix the compensation field on the slope with the opposite detuning
−κ/2, because according to Eq. B.5, δωc0 = −δωc/G(we assume the other term a static shift
The residual dephasing due to ∆N 6= 0 cannot be compensated by the compensation field.
B.3 Noise from fluctuations of total atom number
In practice, we might be more concerned about the fluctuation in δφ0ac due to fluctuations in N (Eq. B.5). As the compensation field can not compensate the fluctuations in φac, we should instead choose zero detuning to be less sensitive to N fluctuation.
The benchmark would be the fluctuation in φac (standard deviation ∆) due to the PSN of the probe:
where hnci= np/2. For the compensation field, at zero detuning, δω0cis first-order insensitive to δN, so we consider the second order:
δn0c≈ −n0p
Note that now as the compensation field is on resonance, the compensation condition is modified to n0p= Gnp/2. Recall Eq.B.9, we have
Now we compare it with PSN of the probe:
We briefly discuss the trapping effect and spontaneous scattering of the compensation field.
Recall the trapping potential
U ∝ Ic
∆c
∝ nc
ωat/2 (B.17)
where Icis the intra-cavity intensity. And the scattering rate Γsc∝ Ic
The concern is more the trapping effect. Our calculation shows that for a realistic probe power, the trapping potential of the compensation field can reach 0.2 recoil energy which is more than 10% of the thermal energy. The real effect on the atoms needs to be verified experimentally.
B.4 Possible scheme
Although I showed that the compensation field cannot compensate the residual light shift from ∆N, a modified scheme can work. Similar to the original composite measurement, we can have two probes with toggled detuning at ±κ/2 respectively. It is the two probes that ensures the total probe photon number always average to np/2, independent of ∆N. The compensate field is at zero detuning to compensate the total light shift. If they are separated in time, the minimum duration would be 3/2 trapping period ∼ 13.5 ms. There are still advantages compared to a composite measurement with a MW π pulse.