Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2611.01205

Pump-scheme dependence of few-emitter lasers: exact multilevel correlations and photon statistics

Stephan Hartmann

quant-phphysics.atom-phphysics.opticsSubmitted Oct 31, 2026• Announced Nov 3, 2026
arXivHTMLPDF

Abstract

Few-emitter laser models usually replace the excitation cycle of the gain medium by a single Markovian pump jump of a two-level emitter. We test this replacement with the cavity and laser-transition parameters held fixed and the two-level pump rate fitted to the same mean photon number, using an exact damping-basis continued-fraction solution of the cavity--emitter master equation for $N$ identical $d$-level emitters with arbitrary independent incoherent transitions and one cavity-coupled transition, in the known reduced space of level populations and lasing coherences. First, a pump-quantum filtration shows that the inter-emitter coherence of $Λ$ and ladder media starts at second rather than first order in the pump; we evaluate its leading coefficient from the exact finite-system series and map the recycling rate at which its sign changes. Second, at matched photon number, rate-limited steps in the pump cycle push the long-time output-counting Fano factor of two and three emitters well below one: for two emitters it reaches 0.66 at a mean cavity occupation of 0.3 and is still 0.74 at an occupation of one, where $g^{(2)}(0)$ is within 3% of unity, because the anticorrelation sits at delays of about half a cavity lifetime rather than at zero delay. For two and three emitters the fitted two-level model gets both statistics wrong and places the output above the shot-noise level. Against independent single-emitter lasers at the same output, the common cavity erodes rather than enhances this single-emitter effect.

Comments
22 pages, 10 figures, 8 tables

Related papers

Exact damping-basis continued fractions for few-atom lasers: collective correlations beyond second-order cumulant closure

Stephan Hartmann

We combine the Briegel--Englert damping basis of a lossy cavity mode with the permutation-invariant Liouville basis of \(N\) identical, incoherently pumped two-level emitters. The resulting Liouvillian is exactly block tridiagonal in a single radial field index, with atomic blocks of dimension \(\binom{N+3}{3}\). The stationary state follows from a matrix continued fraction whose asymptotic closure is a linear matrix equation that we prove to be uniquely solvable throughout the dissipative regime \(A>0\), and the same construction in the coherence-one sector yields the emission spectrum. The method is validated against direct diagonalization, weak-pump perturbation theory, and closed-system spectra, reaches \(N=16\) in tens of seconds in the few-quanta regime, and has a quantified precision limit at large photon number. We use it to derive an exact two-atom photon-balance identity that isolates the cavity-induced pair coherence and relates it to a singlet--triplet population difference; to establish an exact pump-order hierarchy of connected correlations for a zero-temperature cavity and a parity property of the minimal collective-spin sector; and to benchmark the second-order cumulant closure used in few-emitter laser theory, which we find to be semi-quantitative for the photon number but to predict the wrong sign of the inter-emitter coherence above inversion. In a good-cavity regime the exact solution shows near-Poissonian statistics and strong line narrowing for two and three emitters.

quant-pharXiv:2609.17785•Sep 17, 2026•0 citations

Photon statistics of a single atom laser

Bobby D. Jones, Shohini Ghose, James Clemens +2 more

We consider a laser model consisting of a single four-level or three-level atom, an optical cavity, and an incoherent pump. Results for photon statistics for varying pump levels are obtained using a quantum trajectory algorithm. In particular, we calculate the mean photon number, Fano factor (which is the variance over the mean). We examine that the behavior of the single-atom device as \ensuremath{\beta}, the fraction of spontaneous emission into the lasing mode, is varied. Typical values considered for \ensuremath{\beta} are $0.01<\ensuremath{\beta}<1.0.$ We find that for large enough \ensuremath{\beta}, lasing action, with properties similar to those predicted by semiclassical theories that factorize atom-field correlations and use a small-noise approximation, can occur. Squeezing can occur as \ensuremath{\beta} is increased. There is no evidence of a sharp phase transition from weakly excited thermal light to coherent light at a particular pump power. This is consistent with work on many-atom lasers with \ensuremath{\beta} values in the range considered here. As \ensuremath{\beta} is increased, the output goes from quasithermal light to coherent and finally to squeezed light, progressing into a fully quantum-mechanical regime. We also consider the effects of cavity damping and spontaneous emission rates on these results.

Physical Review A•Oct 1, 1999•57 citations

Few emitters in a cavity: from cooperative emission to individualization

Alexia Auffèves, Dario Gerace, S. Portolan +2 more

We study the temporal correlations of the field emitted by an electromagnetic resonator coupled to a mesoscopic number of two-level emitters that are incoherently pumped by a weak external drive. We solve the master equation of the system for increasing number of emitters and as a function of the cavity quality factor, and we identify three main regimes characterized by well-distinguished statistical properties of the emitted radiation. For small cavity decay rates, the emission events are uncorrelated and the number of photons in the emitted field becomes larger than one, resembling the build-up of a laser field inside the cavity. At intermediate decay rates (as compared with the emitter–cavity coupling) and for a few emitters, the statistics of the emitted radiation is bunched and strikingly dependent on the parity of the number of emitters. The latter property is related to the cooperativity of the emitters mediated by their coupling to the cavity mode, and its connection with steady-state subradiance is discussed. Finally, in the bad cavity regime the typical situation of emission from a collection of individual emitters is recovered. We also analyze how the cooperative behavior evolves as a function of pure dephasing, which allows us to recover the case of a classical source made of an ensemble of independent emitters, similar to what is obtained for a very leaky cavity. State-of-the-art techniques of Q -switch of resonant cavities, allied with the recent capability of tuning single emitters in and out of resonance, suggest this system to be a versatile source of different quantum states of light.

arXiv:1106.0836•New Journal of Physics•Sep 9, 2011•91 citations

An analytical approach to collective effects in nanolasers (Conference Presentation)

Emil C. André, Игорь Евгеньевич Проценко, Alexander V. Uskov +2 more

Traditionally, the active material in a laser is modelled as independent emitters, but in recent years it has become increasingly clear that radiative coupling between emitters can significantly change the characteristics of small lasers. Collective effects in free space such as superradiance have been studied extensively [1,2], but the effects of inter-emitter correlation in micro- and nano-cavities need further examination to be put on firm theoretical ground. Several studies of collective effects in nano-cavities have been made [3-6], but the theoretical models employed are intricate, and numerical methods are needed both to generate the dynamic equations and to solve them. We propose a model where the complexity is strongly reduced, allowing analytical solutions [7]. We consider a collection of identical two-level emitters interacting with a single cavity mode. We start from Maxwell-Bloch equations, but instead of making the typical adiabatic elimination of the polarization, we allow the polarization decay rate to be of the same magnitude or smaller than other decay rates. Hence, the traditional laser rate equations for the photon number and the population inversion must be supplemented by equations for the emitter-field correlation and the emitter-emitter correlation. This gives us four generalized laser rate equations, which we solve analytically in steady state. Comparing with the steady state results obtained from the traditional laser rate equations we see that inclusion of collective effects leads to a reduction of the photon number for small pump rates, similarly to what is found in [4]. From the generalized laser rate equations, we derive a measure of the strength of collective effects in terms of laser parameters: This describes the difference between results with and without inter-emitter correlations, and it goes smoothly to zero as we approach parameter values where the traditional laser rate equations become valid. To gain insight into the photon statistics of the laser, we construct dynamic equations for higher order correlations of operators. We derive an analytical expression for the zero-delay photon auto-correlation function, and for low pump rates we find that the interaction of emitters results in super-thermal values of the auto-correlation. This feature is observed in experiments and numerical models [4-5], and with our analytical expressions, we are able to pinpoint the parameter combinations for which the collective effects have the largest impact. Considering the same model in terms of the Fourier components of the operators, we find results for the photon number that agree well with the previous approach, while allowing computation of the linewidth. Thus, we can examine how emitter-emitter correlation affects the line broadening of the laser.

Semiconductor Lasers and Laser Dynamics•May 23, 2018•0 citations

Sub-Poissonian single-emitter laser: Coherent pump vs incoherent

Ларионов Николай Владимирович, Ilya V. Safonov

We theoretically investigate the properties of a single-emitter laser with coherent pump: two-level atom interacting with a single damping cavity mode which is resonantly excited by a coherent field. With the help of numerical simulation of the master equation we show that such pumping mechanism is more preferably than incoherent pump for achieving of sub-Poissonian photon statistics.

Modern Physics Letters A•Jan 16, 2020•1 citation

Properties of nanolasers based on few discrete emitters

Anders Lund, P. Nielsen, Michael Lorke +2 more

The laser has evolved from table size apparatuses to truly nano sized devices, in much the same way that computer chips have been continuously minimized. The few-emitter nanolaser represents an extreme in terms of size. The emitters can be either atoms or quantum dots, and they are coupled to a high-Q optical cavity. This system has previously been studied for the one and two emitters case, see e.g. [1, 2]. The emitters and cavity are modelled as two-level systems and a harmonic oscillator, respectively. The coupled system is modelled using the Jaynes-Cummings Hamiltonian and the two-level systems are pumped incoherently by a rate P. Solutions are found using the corresponding master equation. However, with cavity populations exceeding 100 and several emitters, the dimension of the Hilbert space of the system becomes too large to handle efficiently on a conventional computer. E.g. for four emitters and 100 photon states the density matrix has more than 2.5 × 106 elements. We have been able to simplify the problem significantly by adiabatically eliminating the photon-assisted polarizations and the correlations between emitters and cavity [3]. This results in a set of rate equations for the population of the cavity, na, and the occupation of the emitters. Fig. 1a) shows na for up to 4 emitters coupled to a cavity, as a function of the pumping rate P. The figure also shows results from a full master equation solution, and the correspondence is very good for large values of P. In Fig. 1b) the second order correlation functions g(2)(0) are shown, as obtained using the full model. The second order correlation functions become 1 for sufficiently large values of P signifying the onset of lasing. However, for larger values of P the laser quenches and the emission becomes chaotic (g(2)(0) = 2). This quenching effect is well-known in two-level models [3], but might not be realistic for semiconductor quantum dots. A proposed application for these nanolasers is in on-chip optical interconnects. In this application the modulation bandwidth is important. We have calculated the modulation bandwidth using a small-signal analysis of the rate equations, which has the advantage compared the full model of yielding a semianalytical result. The results are shown in Fig. 1c). The nanolaser has a high bandwidth, which is mostly limited by the cavity lifetime (/ 1/), but shows a prominent drop during the lasing regime. The modulation bandwidth again becomes limited by as the output becomes chaotic.

•Jan 1, 2012•0 citations
Few-emitter laser models usually replace the excitation cycle of the gain medium by a single Markovian pump jump of a two-level emitter. We test this replacement with the cavity and laser-transition parameters held fixed and the two-level pump rate fitted to the same mean photon number, using an exact damping-basis continued-fraction solution of the cavity--emitter master equation for NNN identical ddd-level emitters with arbitrary independent incoherent transitions and one cavity-coupled transition, in the known reduced space of level populations and lasing coherences. First, a pump-quantum filtration shows that the inter-emitter coherence of ΛΛΛ and ladder media starts at second rather than first order in the pump; we evaluate its leading coefficient from the exact finite-system series and map the recycling rate at which its sign changes. Second, at matched photon number, rate-limited steps in the pump cycle push the long-time output-counting Fano factor of two and three emitters well below one: for two emitters it reaches 0.66 at a mean cavity occupation of 0.3 and is still 0.74 at an occupation of one, where g(2)(0)g^{(2)}(0)g(2)(0) is within 3% of unity, because the anticorrelation sits at delays of about half a cavity lifetime rather than at zero delay. For two and three emitters the fitted two-level model gets both statistics wrong and places the output above the shot-noise level. Against independent single-emitter lasers at the same output, the common cavity erodes rather than enhances this single-emitter effect.
We combine the Briegel--Englert damping basis of a lossy cavity mode with the permutation-invariant Liouville basis of NNN identical, incoherently pumped two-level emitters. The resulting Liouvillian is exactly block tridiagonal in a single radial field index, with atomic blocks of dimension (N+33)\binom{N+3}{3}(3N+3​). The stationary state follows from a matrix continued fraction whose asymptotic closure is a linear matrix equation that we prove to be uniquely solvable throughout the dissipative regime A>0A>0A>0, and the same construction in the coherence-one sector yields the emission spectrum. The method is validated against direct diagonalization, weak-pump perturbation theory, and closed-system spectra, reaches N=16N=16N=16 in tens of seconds in the few-quanta regime, and has a quantified precision limit at large photon number. We use it to derive an exact two-atom photon-balance identity that isolates the cavity-induced pair coherence and relates it to a singlet--triplet population difference; to establish an exact pump-order hierarchy of connected correlations for a zero-temperature cavity and a parity property of the minimal collective-spin sector; and to benchmark the second-order cumulant closure used in few-emitter laser theory, which we find to be semi-quantitative for the photon number but to predict the wrong sign of the inter-emitter coherence above inversion. In a good-cavity regime the exact solution shows near-Poissonian statistics and strong line narrowing for two and three emitters.
We consider a laser model consisting of a single four-level or three-level atom, an optical cavity, and an incoherent pump. Results for photon statistics for varying pump levels are obtained using a quantum trajectory algorithm. In particular, we calculate the mean photon number, Fano factor (which is the variance over the mean). We examine that the behavior of the single-atom device as \ensuremath{\beta}, the fraction of spontaneous emission into the lasing mode, is varied. Typical values considered for \ensuremath{\beta} are $0.01<\ensuremath{\beta}<1.0.$ We find that for large enough \ensuremath{\beta}, lasing action, with properties similar to those predicted by semiclassical theories that factorize atom-field correlations and use a small-noise approximation, can occur. Squeezing can occur as \ensuremath{\beta} is increased. There is no evidence of a sharp phase transition from weakly excited thermal light to coherent light at a particular pump power. This is consistent with work on many-atom lasers with \ensuremath{\beta} values in the range considered here. As \ensuremath{\beta} is increased, the output goes from quasithermal light to coherent and finally to squeezed light, progressing into a fully quantum-mechanical regime. We also consider the effects of cavity damping and spontaneous emission rates on these results.
We study the temporal correlations of the field emitted by an electromagnetic resonator coupled to a mesoscopic number of two-level emitters that are incoherently pumped by a weak external drive. We solve the master equation of the system for increasing number of emitters and as a function of the cavity quality factor, and we identify three main regimes characterized by well-distinguished statistical properties of the emitted radiation. For small cavity decay rates, the emission events are uncorrelated and the number of photons in the emitted field becomes larger than one, resembling the build-up of a laser field inside the cavity. At intermediate decay rates (as compared with the emitter–cavity coupling) and for a few emitters, the statistics of the emitted radiation is bunched and strikingly dependent on the parity of the number of emitters. The latter property is related to the cooperativity of the emitters mediated by their coupling to the cavity mode, and its connection with steady-state subradiance is discussed. Finally, in the bad cavity regime the typical situation of emission from a collection of individual emitters is recovered. We also analyze how the cooperative behavior evolves as a function of pure dephasing, which allows us to recover the case of a classical source made of an ensemble of independent emitters, similar to what is obtained for a very leaky cavity. State-of-the-art techniques of Q -switch of resonant cavities, allied with the recent capability of tuning single emitters in and out of resonance, suggest this system to be a versatile source of different quantum states of light.
Traditionally, the active material in a laser is modelled as independent emitters, but in recent years it has become increasingly clear that radiative coupling between emitters can significantly change the characteristics of small lasers. Collective effects in free space such as superradiance have been studied extensively [1,2], but the effects of inter-emitter correlation in micro- and nano-cavities need further examination to be put on firm theoretical ground. Several studies of collective effects in nano-cavities have been made [3-6], but the theoretical models employed are intricate, and numerical methods are needed both to generate the dynamic equations and to solve them. We propose a model where the complexity is strongly reduced, allowing analytical solutions [7]. We consider a collection of identical two-level emitters interacting with a single cavity mode. We start from Maxwell-Bloch equations, but instead of making the typical adiabatic elimination of the polarization, we allow the polarization decay rate to be of the same magnitude or smaller than other decay rates. Hence, the traditional laser rate equations for the photon number and the population inversion must be supplemented by equations for the emitter-field correlation and the emitter-emitter correlation. This gives us four generalized laser rate equations, which we solve analytically in steady state. Comparing with the steady state results obtained from the traditional laser rate equations we see that inclusion of collective effects leads to a reduction of the photon number for small pump rates, similarly to what is found in [4]. From the generalized laser rate equations, we derive a measure of the strength of collective effects in terms of laser parameters: This describes the difference between results with and without inter-emitter correlations, and it goes smoothly to zero as we approach parameter values where the traditional laser rate equations become valid. To gain insight into the photon statistics of the laser, we construct dynamic equations for higher order correlations of operators. We derive an analytical expression for the zero-delay photon auto-correlation function, and for low pump rates we find that the interaction of emitters results in super-thermal values of the auto-correlation. This feature is observed in experiments and numerical models [4-5], and with our analytical expressions, we are able to pinpoint the parameter combinations for which the collective effects have the largest impact. Considering the same model in terms of the Fourier components of the operators, we find results for the photon number that agree well with the previous approach, while allowing computation of the linewidth. Thus, we can examine how emitter-emitter correlation affects the line broadening of the laser.
The laser has evolved from table size apparatuses to truly nano sized devices, in much the same way that computer chips have been continuously minimized. The few-emitter nanolaser represents an extreme in terms of size. The emitters can be either atoms or quantum dots, and they are coupled to a high-Q optical cavity. This system has previously been studied for the one and two emitters case, see e.g. [1, 2]. The emitters and cavity are modelled as two-level systems and a harmonic oscillator, respectively. The coupled system is modelled using the Jaynes-Cummings Hamiltonian and the two-level systems are pumped incoherently by a rate P. Solutions are found using the corresponding master equation. However, with cavity populations exceeding 100 and several emitters, the dimension of the Hilbert space of the system becomes too large to handle efficiently on a conventional computer. E.g. for four emitters and 100 photon states the density matrix has more than 2.5 × 106 elements. We have been able to simplify the problem significantly by adiabatically eliminating the photon-assisted polarizations and the correlations between emitters and cavity [3]. This results in a set of rate equations for the population of the cavity, na, and the occupation of the emitters. Fig. 1a) shows na for up to 4 emitters coupled to a cavity, as a function of the pumping rate P. The figure also shows results from a full master equation solution, and the correspondence is very good for large values of P. In Fig. 1b) the second order correlation functions g(2)(0) are shown, as obtained using the full model. The second order correlation functions become 1 for sufficiently large values of P signifying the onset of lasing. However, for larger values of P the laser quenches and the emission becomes chaotic (g(2)(0) = 2). This quenching effect is well-known in two-level models [3], but might not be realistic for semiconductor quantum dots. A proposed application for these nanolasers is in on-chip optical interconnects. In this application the modulation bandwidth is important. We have calculated the modulation bandwidth using a small-signal analysis of the rate equations, which has the advantage compared the full model of yielding a semianalytical result. The results are shown in Fig. 1c). The nanolaser has a high bandwidth, which is mostly limited by the cavity lifetime (/ 1/), but shows a prominent drop during the lasing regime. The modulation bandwidth again becomes limited by as the output becomes chaotic.