The General Quantum Limit for and the Optimization of the Minimum Measurable Frequency Shift in a Laser

arXiv:2607.10518 · quant-ph · Submitted 2026-07-12 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "The General Quantum Limit for and the Optimization of the Minimum Measurable Frequency Shift in a Laser".

Mira: Here's the gist of the paper. The gist The minimum measurable frequency shift (MMFS) for a single-mode ideal laser is determined by a combination of phase diffusion caused by spontaneous emission and shot noise caused by the vacuum mode, which can be much larger than unity under certain conditions.

Kai: General Model for the MMFS The MMFS is determined by two independent effects: the phase diffusion caused by SE into the lasing mode, and the shot-noise caused by the VM accompanying a coherent state.

Mira: The overall MMFS can be expressed as a combination of these two effects in Eqn.

Kai: (18).

Paper summary: Kai: So we just talked about how the paper introduces this combination of phase diffusion from spontaneous emission and shot noise from the vacuum mode, and how it leads to this geometric mean formula for the MMFS. Now, let's look at what they claim is important about this work.

Mira: They go on to define these two effects separately: spontaneous emission causing phase diffusion, and the vacuum mode causing shot noise uncertainty in photon number.

Lev: But from an engineering standpoint, how does a theorist translate that into something we can actually design? Does it just give us a theoretical floor or something more practical?

Kai: They show that the rate of spontaneous emission into the lasing mode is related to a single photon in the vacuum mode, which directly leads to phase diffusion for the laser.

Mira: And they link this to the spectral width of the laser, which is called the Schawlow-Townes linewidth. They give an expression for it as one over two STL LC gamma tau gamma equals n >--- Page three --- <ref:2607.10518#pg3>

Lev: So if we think about running this on real hardware, like a superconducting circuit, does that formula rely on assumptions about the cavity lifetime or something we have to worry about?

Kai: The paper says this result is strictly valid when S STL tau tau =, which they say is typically the case for any system of practical interest.

Mira: Okay, so what about the other side of it, the vacuum mode effect? They give a specific expression for that part too.

Lev: What's in it for us when we look at minimizing that shot-noise contribution? Is there a condition they point to where that term becomes negligible?

Kai: They say the overall MMFS is the sum of these two effects, and this minimum value occurs when X equals one which corresponds to two D STL tau tau =, for a certain decay time <ref:2607.10518#pg1>.

Mira: And at that specific delay time, they claim the MMFS due to the vacuum mode becomes negligible compared to Gamma zero.

Lev: So if we were designing a sensor based on this, does it mean we can get much smaller shifts than what was previously thought possible?

Conclusion: Kai: So wrapping up this look at "The General Quantum Limit for and the Optimization of the Minimum Measurable Frequency Shift in a Laser," they’ve essentially shown that the MMFS isn't just one single limitation, but this interplay between spontaneous emission and vacuum noise.

Mira: The authors determined the optimal MMFS value for three specific sensing modalities: an unbalanced Mach-Zehnder Interferometer, a passive Fabry-Perot cavity, and heterodyning with a reference laser.

Lev: What's the big picture here? Does this change how we think about precision measurement in general? Is it just more complex math or something that impacts hardware design?

Kai: The implication is that for practical sensors, the MMFS is determined by this geometric mean of the measurement bandwidth and the Schawlow-Townes linewidth, multiplied by a factor that can be much larger than one.

Mira: And they found specific conditions needed to reach these optimal values for those three different setups.

Lev: So what does this mean for someone who just listens to our show? It means precision is limited by the interplay of fundamental quantum noise sources, and you have to optimize your setup to get the best possible result.

Northwestern University

quant-ph

Submitted: 2026-07-12

Updated: 2026-10-21

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 72/100

The gist: The gist The minimum measurable frequency shift (MMFS) for a single-mode ideal laser is determined by a combination of phase diffusion caused by spontaneous emission and shot noise caused by the

Key concepts

Minimum Measurable Frequency Shift (MMFS): This is the smallest detectable change in the laser's frequency that can be measured by a sensor. It is limited by two main physical phenomena: phase diffusion caused by spontaneous emission and shot noise arising from vacuum fluctuations, which together set a fundamental lower bound on measurement precision.
Phase Diffusion (SE): This effect arises from spontaneous emission where photons are randomly emitted into the lasing mode. This random process causes the laser's phase to drift over time, effectively broadening the spectral linewidth of the laser and limiting how precisely its frequency can be measured.
Shot Noise (VM): Shot noise is a form of uncertainty in photon number within a coherent state, directly caused by vacuum fluctuations. This noise contributes to the MMFS and can be minimized through specific measurement conditions, such as adjusting the delay time or operating point, to improve sensitivity.

Summary

The gist The minimum measurable frequency shift (MMFS) for a single-mode ideal laser is determined by a combination of phase diffusion caused by spontaneous emission and shot noise caused by the vacuum mode, which can be much larger than unity under certain conditions.

General Model for the MMFS

The MMFS is determined by two independent effects: the phase diffusion caused by SE into the lasing mode, and the shot-noise caused by the VM accompanying a coherent state. The overall MMFS can be expressed as a combination of these two effects in Eqn. (18).

Effect of Spontaneous Emission (SE)

The rate of spontaneous emission into the lasing mode is related to the single photon in the vacuum mode, leading to phase diffusion for the laser. The spectral width (Schawlow-Townes Linewidth) of the laser is given by 1/2 STL LC γ τγ = n. The MMFS due to the effect of SE only is given by Eqn. (2). This result is strictly valid when S STL ττ =, which is typically the case for any system of practical interest.

Effect of the Vacuum Mode (VM)

The uncertainty in the number of photons in a coherent state is called shot-noise and is directly attributable to the vacuum mode. The MMFS due to effect of the VM only is found to be given by Eqn. (5). Optimal measurement conditions can be used to reach this limit for ∆ω MIN VM, as shown in Section 3.1.

Overall MMFS and Optimization

The overall MMFS is given by the sum of the effects due to SE and VM in Eqn. (18). The minimum value of this quantity occurs at X = 1, which corresponds to 2 D STL ττ =, for a certain decay time. For this optimal choice of the delay time, the MMFS due to the VM becomes negligible compared to Γ0.

Modality-Specific Minimization

The paper determines the optimal value for the MMFS for three different sensing modalities:

Unbalanced Mach-Zehnder Interferometer (UMZI): The optimal operating point is at where the fringe has the highest slope, corresponding to 0 φ π = ± / 2.
Passive Fabry-Perot Cavity (FPC): The minimum value of the VM contribution to the MMFS for the FPC occurs at ρ = 2 as R → 1.
Heterodyning with a reference laser: The smallest value of the MMFS in Eqn. (33) occurs at X = 0.69, which corresponds to 1.38 S STL ττ =, and this is consistent with the optimal duration for a single measurement time.

The final overall MMFS for the FPC is given approximately by ∆ Γ ω MIN SE, 0. This result shows that by making an FPC with a linewidth much narrower than the linewidth of the cavity used for making the laser, it is possible to obtain an MMFS much smaller than what Dorschner et al. claimed to be the standard limit.

The paper concludes that for practical sensors, the MMFS is found to be given by a geometric mean of the measurement bandwidth and the Schawlow-Townes Linewidth, multiplied by a factor which can be much larger than unity under certain conditions. The authors determine the optimal value for the MMFS for three different sensing modalities, an unbalanced MachZehnder Interferometer, a passive Fabry-Perot cavity (FPC), and heterodyning with a reference laser, and identify the conditions needed for reaching the optimal value.

The paper concludes that for the FPC, the value of ∆ω MIN SE, remains close to Γ0 when the value of ∆ω MIN VM, is minimized. Thus, under optimal operating conditions, the overall MMFS for the FPC is given approximately by ∆ Γ ω MIN SE, 0. It should also be noted that this is also the limit for measuring the change in the resonance frequency of an FPC.

The paper concludes that for heterodyning with a reference laser, under optimal operating conditions, the overall MMFS for the heterodyning scheme is given approximately by ∆ Γ ω MIN SE, 0. This result shows that while operating at 2 S STL ττ = drastically reduced the value of the MMFS due to the VM, it does not correspond to any significant increase in the value of the MMFS due to the SE. The paper concludes that for heterodyning with a reference laser, under optimal operating conditions, the overall MMFS for the heterodyning scheme is given approximately by ∆ Γ ω MIN SE, 0. It should also be noted that this is also the limit for measuring the change in the resonance frequency of an FPC.

The paper concludes that for all practical sensors, the MMFS is found to be given by a geometric mean of the measurement bandwidth and the Schawlow-Townes Linewidth, multiplied by a factor which can be much larger than unity under certain conditions. The authors determine the optimal value for the MMFS for three different sensing modalities, an unbalanced MachZehnder Interferometer, a passive Fabry-Perot cavity (FPC), and heterodyning with a reference laser, and identify the conditions needed for reaching the optimal value.

Improvements for AI systems

Improved frequency shift sensing for gyroscopes by optimizing measurement time duration based on phase diffusion and shot noise. The system can reach a minimum measurable frequency shift determined by a combination of phase diffusion caused by SE into the lasing mode, and the shot-noise caused by the vacuum mode (VM) that accompanies a coherent state.
Enhanced performance for unbalanced interferometers by employing hopping techniques to minimize vacuum mode contribution. The improved system can reduce the MMFS due to VM by a factor of 2 compared to measurements at one side only, leading to an optimal value given by MIN VM M S S o LC M STL when operating at the point where the slope of the fringe is maximum.
Superior frequency shift measurement using passive Fabry Perot cavities by tuning the cavity linewidth relative to the laser linewidth. The improved system can obtain an MMFS much smaller than claimed standard limits by making an FPC with a linewidth much narrower than the linewidth of the cavity used for making the laser, leading to a minimum VM contribution of approximately 0 1.4 STL under optimal conditions.
Optimized heterodyne detection for frequency shift using a reference laser by selecting the optimal spectral analyzer bandwidth. The improved system can minimize the VM contribution to an overall MMFS by choosing a measurement point where the smallest value of the MMFS in Eqn (33) occurs at X 0.69, which corresponds to 1.38 STL.
Quantum limit determination using Cramer-Rao bounds for unbalanced Mach-Zehnder Interferometers. The improved system can precisely determine the variance of a frequency shift parameter by calculating the Quantum CRB based on the Fisher Information derived from a quantum mechanical description, which converges to rigorous results in the limit where c STL τ τ = and N 1.