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arXiv:quant-ph/9903082

The ultimate quantum limit to the linewidth of lasers

Howard M. Wiseman

Published Mar 25, 1999• 0 citations
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Abstract

The standard quantum limit to the linewidth of a laser for which the gain medium can be adiabatically eliminated is $\ell_{0}=κ/2\bar{n}$. Here $κ$ is the intensity damping rate and $\bar{n}$ the mean photon number. This contains equal contributions from the loss and gain processes, so that simple arguments which attribute the linewidth wholly to phase noise from spontaneous gain are wrong. I show that an {\em unstimulated} gain process actually introduces no phase noise, so that the ultimate quantum limit to the laser linewidth comes from the loss alone and is equal to $\ell_{ult}= κ/4\bar{n}$. I investigate a number of physical gain mechanisms which attempt to achieve gain without phase noise: a linear atom-field coupling with finite interaction time; a nonlinear atom-field coupling; and adiabatic photon transfer using a counterintuitive pulse sequence. The first at best reaches the standard limit $\ell_{0}$, the second reaches $3/4 \ell_{0}$, while the third reaches the ultimate limit of $\ell_{ult} = 1/2 \ell_{0}$.

DOI
10.48550/arxiv.quant-ph/9903082

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The standard quantum limit to the linewidth of a laser for which the gain medium can be adiabatically eliminated is ℓ0=κ/2nˉ\ell_{0}=κ/2\bar{n}ℓ0​=κ/2nˉ. Here κκκ is the intensity damping rate and nˉ\bar{n}nˉ the mean photon number. This contains equal contributions from the loss and gain processes, so that simple arguments which attribute the linewidth wholly to phase noise from spontaneous gain are wrong. I show that an {\em unstimulated} gain process actually introduces no phase noise, so that the ultimate quantum limit to the laser linewidth comes from the loss alone and is equal to ℓult=κ/4nˉ\ell_{ult}= κ/4\bar{n}ℓult​=κ/4nˉ. I investigate a number of physical gain mechanisms which attempt to achieve gain without phase noise: a linear atom-field coupling with finite interaction time; a nonlinear atom-field coupling; and adiabatic photon transfer using a counterintuitive pulse sequence. The first at best reaches the standard limit ℓ0\ell_{0}ℓ0​, the second reaches 3/4ℓ03/4 \ell_{0}3/4ℓ0​, while the third reaches the ultimate limit of ℓult=1/2ℓ0\ell_{ult} = 1/2 \ell_{0}ℓult​=1/2ℓ0​.
The standard quantum limit to the linewidth of a laser for which the gain medium can be adiabatically eliminated is ${l}_{0}=\ensuremath{\kappa}/2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ Here $\ensuremath{\kappa}$ is the intensity damping rate and $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ the mean photon number. This contains equal contributions from the loss and gain processes, so that simple arguments which attribute the linewidth wholly to phase noise from spontaneous gain are wrong. I show that an unstimulated gain process actually introduces no phase noise, so that the ultimate quantum limit to the linewidth comes from the loss alone and is equal to ${l}_{\mathrm{ult}}=\ensuremath{\kappa}/4\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ I investigate a number of physical gain mechanisms which attempt to achieve gain without phase noise: a linear atom-field coupling with a finite interaction time, a nonlinear atom-field coupling, and adiabatic photon transfer using a counterintuitive pulse sequence. The first at best reaches the standard limit l0,{l}_{0},l0​, the second reaches 34l0,\frac{3}{4}{l}_{0},43​l0​, and the third reaches the ultimate limit of lult=12l0.{l}_{\mathrm{ult}}=\frac{1}{2}{l}_{0}.lult​=21​l0​.
Due to their high coherence, Lasers are a ubiquitous tool in science. The standard quantum limit for the phase coherence time was first introduced by [A. Schawlow and C. Townes, Phys. Rev. 112, 1940 (1958)], who showed that the minimum possible laser linewidth is determined by the linewidth of the laser cavity divided by twice the number of photons in the cavity. Later, Wiseman showed theoretically that by using Susskind-Glogower (SG) operators to couple the gain medium to the laser cavity it is possible to eliminate pump noise, but not loss noise. This decreases the minimum laser linewidth, though only by a factor of two. In this article, we show that by engineering the coupling between the laser cavity and the output port it is possible to eliminate most of the loss noise as well and construct a laser that has a vastly narrower linewidth, narrower than the standard quantum limit by a factor equal to the number of photons in the laser cavity. We establish a roadmap for building such a device in the laboratory by using Josephson junctions and linear circuit elements to build coupling circuits that behave like SG operators for a range of cavity photon occupancies and using them to couple the laser cavity to both the gain medium and the output port. This device could be an ultra-coherent, cryogenic light source for microwave quantum information experiments. Further, our laser provides highly squeezed light and could be modified to provide designer quantum light which is an important resource for CV/linear optical quantum computing, readout of quantum states in superconducting quantum computers, quantum metrology, and quantum communication. Finally, our proposal relies on the tools and elements of superconducting quantum information, and thus is a clear example of how quantum engineering techniques can inspire us to re-imagine the limits of conventional quantum systems such as the laser.
The problem of the quantum-limited or intrinsic linewidth of a good-cavity laser is revisited. Starting from the Scully-Lamb master equation, we present a fully analytical treatment to determine the correlation function and the spectrum of the cavity field at steady state. For this purpose, we develop an analytical approximation method that implicitly incorporates the microscopic fluctuations of both the phase and intensity of the field, and, in addition, takes full account of the saturation of the nonlinear gain. Our main result is a simple formula for the quantum-limited linewidth that is valid from near to far above threshold and also includes the presence of thermal photons. Close to the threshold, the linewidth is twice as large as predicted by the standard phase-diffusion treatment neglecting intensity fluctuations, and even 50% above threshold the increase is still considerable. In general, quantum fluctuations of the intensity are present and continue to influence the linewidth as long as the photon-number distribution is not strictly Poissonian. This inherent relationship is displayed by a formula relating the linewidth and the Mandel Q parameter. More than 100% above treshold the linewidth is found to be smaller than predicted by the standard treatment, since the simple phase-diffusion model increasingly overestimates the rate of phase fluctuations by neglecting gain saturation. In the limit of a very large mean photon number the expected perfectly coherent classical field is obtained.
Due to their high coherence, lasers are ubiquitous tools in science. We show that by engineering the coupling between the gain medium and the laser cavity as well as the laser cavity and the output port, it is possible to eliminate most of the noise due to photons entering as well as leaving the laser cavity. Hence, it is possible to reduce the laser linewidth by a factor equal to the number of photons in the laser cavity below the standard quantum limit. We design and theoretically analyze a superconducting circuit that uses Josephson junctions, capacitors and inductors to implement a microwave laser, including the low-noise couplers that allow the design to surpass the standard quantum limit. Our proposal relies on the elements of superconducting quantum information, and thus is an example of how quantum engineering techniques can inspire us to re-imagine the limits of conventional quantum systems.
We use standard quantum laser theory to examine the linewidth of lasers with a four-level atomic gain medium as a function of the spontaneous emission rate into the lasing mode and the decay rates from the upper and lower lasing levels. We find that the laser linewidth ceases to exhibit the customary decrease with increased photon number when a critical photon number, proportional to the ratio of the decay rate from the lower lasing level to the rate of spontaneous emission into the lasing mode, is reached. This critical photon number would decrease as the size of the optical cavity is decreased. In addition we show that, in general, the laser linewidth depends critically on the ratio of the overall decay rates from the upper and lower lasing levels.
The quantum-limited linewidth for the Scully-Lamb laser model is discussed in density matrix approach with emphasis on a relation to the field intensity. It is shown that a power-independent term in the linewidth, i.e., the upper-limit of the phase diffusion time, exists due to gain saturation. The linewidth is constant on the field intensity far above threshold, where the photon statistics is similar to the Poisson distribution. The well-known inverse proportionality of the linewidth to the field intensity is valid only in the limited region slightly above threshold, where the photon statistics is quite different from the Poisson distribution. Exact expression of the spontaneous emission rate obtained from the linewidth shows that spontaneous emissions should be attributed to both the gain and the loss mechanisms.