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

The Journal of Physical Chemistry A

On the Interplay of Control Fields and Spontaneous Emission in Laser Cooling

David J. Tannor, Allon Bartana

Published Dec 1, 1999• 83 citations
Publisher

Abstract

Optimal control theory (OCT) is applied to the problem of cooling molecular rotations. The optimal field gives rise to a striking behavior, in which there is no noticeable increase in the lowest rotational state population until the last percent or so of the control interval, at which point the population jumps to 1. Further analysis of the intermediate time interval reveals that cooling is taking place all along, in the sense that the purity of the system, as measured by Tr(ρ 2 ), is increasing monotonically in time. Once the system becomes almost completely pure, the external control field can transfer the amplitude to the lowest rotational state by a completely Hamiltonian manipulation. This mechanism is interesting because it suggests a possible way of accelerating cooling, by exploiting the cooling induced by spontaneous emission to all the ground electronic state levels, not just the lowest rotational level. However, it also raises a major paradox: it may be shown that external control fields, no matter how complicated, cannot change the value of Tr(ρ 2 ); changing this quantity requires spontaneous emission which is inherently uncontrollable. What place is there then for control, let alone optimal control, using external fields? We discuss the resolution to this paradox with a detailed analysis of cooling in a two-level system.

DOI
10.1021/jp992544x

Related papers

Laser cooling of internal degrees of freedom of molecules by dynamically trapped states

David J. Tannor, Ronnie Kosloff, Alon Bartana

In the last several years we have discovered a variety of remarkable pulse strategies for manipulating molecular motion by employing a design strategy we call “local optimization.'' Here we review the concept of local optimization and contrast it with optimal control theory. By way of background, we give highlights from two recent examples of the method: (1) a strategy for eliminating population transfer to one or many excited electronic states during strong field excitation, an effect we call ‘optical paralysis’; (2) a generalization of the counterintuitive STIRAP (stimulated Raman adiabatic passage) pulse sequence from three levels to N levels, a strategy we call ‘straddling STIRAP.' We then turn to a third example, which is the main subject of this paper: laser cooling of molecular internal degrees of freedom. We study a model that includes both coherent interaction with the radiation field and spontaneous emission; the latter is necessary to carry away the entropy from the molecule. An optimal control calculation was performed first and succeeded in producing vibrational cooling, but the resulting pulse sequence was difficult to interpret. Local optimization subsequently revealed the cooling mechanism: the instantaneous phase of the laser is locked to the phase of the transition dipole moment between the excited state amplitude and v=0 of the ground state. Thus, the molecules that reach v=0 by spontaneous emission become decoupled from the field, and no longer absorb, while molecules in all other states are continually repumped. The mechanism could be called “vibrationally selective coherent population trapping,'' in analogy to the corresponding mechanism of velocity selective coherent population trapping in atoms for sub-Doppler cooling of translations.

Faraday Discussions•Jan 1, 1999•72 citations

Cooling molecular vibrations with shaped laser pulses: optimal control theory exploiting the timescale separation between coherent excitation and spontaneous emission

Daniel M. Reich, Christiane P. Koch

Laser cooling of molecules employing broadband optical pumping involves a timescale separation between laser excitation and spontaneous emission. Here, we optimize the optical pumping step using shaped laser pulses. We derive two optimization functionals to drive population into those excited state levels that have the largest spontaneous emission rates to the target state. We show that, when using optimal control, laser cooling of molecules works even if the Franck–Condon map governing the transitions is preferential to heating rather than cooling. Our optimization functional is also applicable to the laser cooling of other degrees of freedom provided the cooling cycle consists of coherent excitation and dissipative de-excitation steps whose timescales are separated.

arXiv:1308.0803•New Journal of Physics•Dec 20, 2013•28 citations

Laser cooling of internal degrees of freedom. II

Allon Bartana, Ronnie Kosloff, David J. Tannor

Theoretical progress in the cooling of internal degrees of freedom of molecules using shaped laser pulses is reported. The emphasis is on general concepts and universal constraints. Several alternative definitions of cooling are considered, including reduction of the von Neumann entropy, −tr{ρ̂logρ̂} and increase of the Renyi entropy, tr{ρ̂2}. A distinction between intensive and extensive considerations is used to analyse the cooling process in open systems. It is shown that the Renyi entropy increase is consistent with an increase in the system phase space density and an increase in the absolute population in the ground state. The limitations on cooling processes imposed by Hamiltonian generated unitary transformations are analyzed. For a single mode system with a ground and excited electronic surfaces driven by an external field it is shown that it is impossible to increase the ground state population beyond its initial value. A numerical example based on optimal control theory demonstrates this result. For this model only intensive cooling is possible which can be classified as evaporative cooling. To overcome this constraint, a single bath degree of freedom is added to the model. This allows a heat pump mechanism in which entropy is pumped by the radiation from the primary degree of freedom to the bath mode, resulting in extensive cooling.

The Journal of Chemical Physics•Jan 22, 1997•128 citations

Laser Control of Collective Spontaneous Emission

Mihai Macovei, Christoph H. Keitel

The collective spontaneous emission of a pair of two coupled three-level radiators in vacuum is investigated in the presence of a possibly intense laser field. The parameters describing the collective interaction along with the population and decay rates of all involved dressed states are shown to be controllable by the applied laser field. In particular, all populations of the collective system may be transferred at will in a reversible way into a subradiant state, allowing effective storage and manipulation of the quantum system.

Physical Review Letters•Sep 16, 2003•46 citations

Coherent control of molecular processes application to cooling internal degrees of freedom

Ronnie Kosloff

Optimal control theory is applied to laser cooling of molecules. The objective is to cool vibrations, using shaped pulses synchronized with the spontaneous emission. The optimal mechanism is found to operate by a “vibrationally selective coherent population trapping”. The trapping condition is that the instantaneous phase of the laser is locked to the phase of the transition dipole moment of v=0 with the excited population. The molecules that reach v=0 by spontaneous emission are then trapped, while the others are continually repumped.

AIP conference proceedings•Jan 1, 2000•0 citations

Optimal control of quantum dissipative dynamics: Analytic solution for cooling the three-level Λ system

Shlomo E. Sklarz, David J. Tannor, Navin Khaneja

We study the problem of optimal control of dissipative quantum dynamics. Although under most circumstances dissipation leads to an increase in entropy (or a decrease in purity) of the system, there is an important class of problems for which dissipation with external control can decrease the entropy (or increase the purity) of the system. An important example is laser cooling. In such systems, there is an interplay of the Hamiltonian part of the dynamics, which is controllable, and the dissipative part of the dynamics, which is uncontrollable. The strategy is to control the Hamiltonian portion of the evolution in such a way that the dissipation causes the purity of the system to increase rather than decrease. The goal of this paper is to find the strategy that leads to maximal purity at the final time. Under the assumption that Hamiltonian control is complete and arbitrarily fast, we provide a general framework by which to calculate optimal cooling strategies. These assumptions lead to a great simplification, in which the control problem can be reformulated in terms of the spectrum of eigenvalues of $\ensuremath{\rho}$, rather than $\ensuremath{\rho}$ itself. By combining this formulation with the Hamilton-Jacobi-Bellman theorem we are able to obtain an equation for the globally optimal cooling strategy in terms of the spectrum of the density matrix. For the three-level $\ensuremath{\Lambda}$ system, we provide a complete analytic solution for the optimal cooling strategy. For this system it is found that the optimal strategy does not exploit system coherences and is a ``greedy'' strategy, in which the purity is increased maximally at each instant.

arXiv:quant-ph/0402143•Physical Review A•May 13, 2004•75 citations
Optimal control theory (OCT) is applied to the problem of cooling molecular rotations. The optimal field gives rise to a striking behavior, in which there is no noticeable increase in the lowest rotational state population until the last percent or so of the control interval, at which point the population jumps to 1. Further analysis of the intermediate time interval reveals that cooling is taking place all along, in the sense that the purity of the system, as measured by Tr(ρ 2 ), is increasing monotonically in time. Once the system becomes almost completely pure, the external control field can transfer the amplitude to the lowest rotational state by a completely Hamiltonian manipulation. This mechanism is interesting because it suggests a possible way of accelerating cooling, by exploiting the cooling induced by spontaneous emission to all the ground electronic state levels, not just the lowest rotational level. However, it also raises a major paradox: it may be shown that external control fields, no matter how complicated, cannot change the value of Tr(ρ 2 ); changing this quantity requires spontaneous emission which is inherently uncontrollable. What place is there then for control, let alone optimal control, using external fields? We discuss the resolution to this paradox with a detailed analysis of cooling in a two-level system.
In the last several years we have discovered a variety of remarkable pulse strategies for manipulating molecular motion by employing a design strategy we call “local optimization.'' Here we review the concept of local optimization and contrast it with optimal control theory. By way of background, we give highlights from two recent examples of the method: (1) a strategy for eliminating population transfer to one or many excited electronic states during strong field excitation, an effect we call ‘optical paralysis’; (2) a generalization of the counterintuitive STIRAP (stimulated Raman adiabatic passage) pulse sequence from three levels to N levels, a strategy we call ‘straddling STIRAP.' We then turn to a third example, which is the main subject of this paper: laser cooling of molecular internal degrees of freedom. We study a model that includes both coherent interaction with the radiation field and spontaneous emission; the latter is necessary to carry away the entropy from the molecule. An optimal control calculation was performed first and succeeded in producing vibrational cooling, but the resulting pulse sequence was difficult to interpret. Local optimization subsequently revealed the cooling mechanism: the instantaneous phase of the laser is locked to the phase of the transition dipole moment between the excited state amplitude and v=0 of the ground state. Thus, the molecules that reach v=0 by spontaneous emission become decoupled from the field, and no longer absorb, while molecules in all other states are continually repumped. The mechanism could be called “vibrationally selective coherent population trapping,'' in analogy to the corresponding mechanism of velocity selective coherent population trapping in atoms for sub-Doppler cooling of translations.
Laser cooling of molecules employing broadband optical pumping involves a timescale separation between laser excitation and spontaneous emission. Here, we optimize the optical pumping step using shaped laser pulses. We derive two optimization functionals to drive population into those excited state levels that have the largest spontaneous emission rates to the target state. We show that, when using optimal control, laser cooling of molecules works even if the Franck–Condon map governing the transitions is preferential to heating rather than cooling. Our optimization functional is also applicable to the laser cooling of other degrees of freedom provided the cooling cycle consists of coherent excitation and dissipative de-excitation steps whose timescales are separated.
Theoretical progress in the cooling of internal degrees of freedom of molecules using shaped laser pulses is reported. The emphasis is on general concepts and universal constraints. Several alternative definitions of cooling are considered, including reduction of the von Neumann entropy, −tr{ρ̂logρ̂} and increase of the Renyi entropy, tr{ρ̂2}. A distinction between intensive and extensive considerations is used to analyse the cooling process in open systems. It is shown that the Renyi entropy increase is consistent with an increase in the system phase space density and an increase in the absolute population in the ground state. The limitations on cooling processes imposed by Hamiltonian generated unitary transformations are analyzed. For a single mode system with a ground and excited electronic surfaces driven by an external field it is shown that it is impossible to increase the ground state population beyond its initial value. A numerical example based on optimal control theory demonstrates this result. For this model only intensive cooling is possible which can be classified as evaporative cooling. To overcome this constraint, a single bath degree of freedom is added to the model. This allows a heat pump mechanism in which entropy is pumped by the radiation from the primary degree of freedom to the bath mode, resulting in extensive cooling.
The collective spontaneous emission of a pair of two coupled three-level radiators in vacuum is investigated in the presence of a possibly intense laser field. The parameters describing the collective interaction along with the population and decay rates of all involved dressed states are shown to be controllable by the applied laser field. In particular, all populations of the collective system may be transferred at will in a reversible way into a subradiant state, allowing effective storage and manipulation of the quantum system.
Optimal control theory is applied to laser cooling of molecules. The objective is to cool vibrations, using shaped pulses synchronized with the spontaneous emission. The optimal mechanism is found to operate by a “vibrationally selective coherent population trapping”. The trapping condition is that the instantaneous phase of the laser is locked to the phase of the transition dipole moment of v=0 with the excited population. The molecules that reach v=0 by spontaneous emission are then trapped, while the others are continually repumped.
We study the problem of optimal control of dissipative quantum dynamics. Although under most circumstances dissipation leads to an increase in entropy (or a decrease in purity) of the system, there is an important class of problems for which dissipation with external control can decrease the entropy (or increase the purity) of the system. An important example is laser cooling. In such systems, there is an interplay of the Hamiltonian part of the dynamics, which is controllable, and the dissipative part of the dynamics, which is uncontrollable. The strategy is to control the Hamiltonian portion of the evolution in such a way that the dissipation causes the purity of the system to increase rather than decrease. The goal of this paper is to find the strategy that leads to maximal purity at the final time. Under the assumption that Hamiltonian control is complete and arbitrarily fast, we provide a general framework by which to calculate optimal cooling strategies. These assumptions lead to a great simplification, in which the control problem can be reformulated in terms of the spectrum of eigenvalues of $\ensuremath{\rho}$, rather than $\ensuremath{\rho}$ itself. By combining this formulation with the Hamilton-Jacobi-Bellman theorem we are able to obtain an equation for the globally optimal cooling strategy in terms of the spectrum of the density matrix. For the three-level $\ensuremath{\Lambda}$ system, we provide a complete analytic solution for the optimal cooling strategy. For this system it is found that the optimal strategy does not exploit system coherences and is a ``greedy'' strategy, in which the purity is increased maximally at each instant.