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Physical Review A

Light amplification without stimulated emission: Beyond the standard quantum limit to the laser linewidth

Howard M. Wiseman

Published Nov 1, 1999• 39 citations
Publisher

Abstract

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 ${l}_{0},$ the second reaches $\frac{3}{4}{l}_{0},$ and the third reaches the ultimate limit of ${l}_{\mathrm{ult}}=\frac{1}{2}{l}_{0}.$

DOI
10.1103/physreva.60.4083

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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=\ensuremathκ/2n\ifmmodeˉ\elseˉ\fi.{l}_{0}=\ensuremath{\kappa}/2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.l0​=\ensuremathκ/2n\ifmmodeˉ\elseˉ\fi. Here \ensuremathκ\ensuremath{\kappa}\ensuremathκ is the intensity damping rate and n\ifmmodeˉ\elseˉ\fi\mathrm{n\ifmmode \bar{}\else \={}\fi{}}n\ifmmodeˉ\elseˉ\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 lult=\ensuremathκ/4n\ifmmodeˉ\elseˉ\fi.{l}_{\mathrm{ult}}=\ensuremath{\kappa}/4\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.lult​=\ensuremathκ/4n\ifmmodeˉ\elseˉ\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​.
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​.
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.
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 show that a saturable amplifier can beat the "3 dB" quantum noise figure of linear amplifiers, approaching the ideal "0 dB" Heisenberg limit. Noise suppression is due to nonlinear saturation effects which cut off the high-intensity tail of the photon distribution. A Monte Carlo simulation of an actual semiconductor laser amplifier leads to a minimum noise figure of 0.51 dB, with a still sizable gain of 10.8 dB. The best performance corresponds to an output signal photon number which equals 0.7 times the saturation number.
We demonstrate that the intensity fluctuations of a laser with a Raman gain medium, described by a large number of independent 3-level atoms, can drop well below the shot noise level. In the limit of a good cavity, where one can adiabatically eliminate the atoms, we find an optimum Mandel Q parameter of Q ≈ −1/2 over a large range of parameters, when the laser is operated well above threshold. This leads to a strong suppression of the low-frequency quantum fluctuations in the laser output. In addition we find that due to suppression of spontaneous emission into the lasing mode, the spectral linewidth of the emitted light can be reduced below the Schawlow-Townes limit by more than one order of magnitude.
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.