Quantum molecular dynamics study of the ammonia inversion transition under pressure
Quantum molecular dynamics study of the pressure dependence of 
the ammonia inversion transition 
I.M. Herbauts and D.J. Dunstan, 
Physics Department, Queen Mary, University of London,  
London E1 4NS, England.  
Abstract:  The mechanism of the shift, broadening and quenching of the ammonia 
inversion frequency with gas pressure has been a problem of lively interest for over 
seventy years. A simple quantum model of the ammonia molecule perturbed by 
collisions with ideal gas molecules displays the essential features of the experimental 
data for NH3 and for ND3. The model does not display the behaviour expected from 
theories of quantum localisation such as quantum state diffusion and decoherence. On 
the other hand, models of perturbed classical oscillators do display similar behaviour 
to our model. The quenching of the ammonia inversion transition cannot therefore be 
interpreted as spatial localisation of the wavefunction.  
Introduction:⎯Since the early days of microwave spectroscopy, the inversion 
transition of the ammonia molecule NH3 and ND3 has been extensively studied 
experimentally and theoretically. Much of the interest lies in the fact that it is the 
smallest and simplest of the pyramidal and enantiomorphic molecules whose ground 
and excited energy eigenstates are the quantum superpositions of two different spatial 
configurations, and that it is light enough that the transitions between the energy 
eigenstates are fast enough to be experimentally accessible. The ammonia molecule 
has two spatial eigenstates L  and R  with the nitrogen atom on one side or the 
other of the plane of hydrogen atoms, and its energy ground and first excited states 
0  and 1  are the symmetric and antisymmetric quantum superpositions of the 
spatial eigenstates (ignoring rotational and vibrational states).   
The ammonia maser is based upon the transition between the energy 
eigenstates, which may also be described as the Rabi oscillation between the spatial 
eigenstates.  However, the inversion transition is seen only at low gas pressure. As the 
gas pressure is increased, the transition broadens, shifts to lower frequency and then 
quenches (the frequency goes to zero).  The ammonia molecule appears to undergo 
spatial localisation as a result of interaction with the environment. This would be of 
immense theoretical interest. In chemistry and in the classical world generally, 
enantiomorphic molecules with distinguishable spatial eigenstates L  and R  are 
always found in their spatial eigenstates (classical behaviour) rather than their energy 
ground states (quantum behaviour).3  Whilst ammonia is not enantiomorphic, it does 
appear to show both behaviours, quantum at low pressure and classical at high 
pressure, if the quenching is considered to be a direct observation of localisation or 
collapse of the wave-function into a spatial eigenstate.  Within the context of the 
decoherence programme, it has been treated quantitatively in that way.4   
In this paper, we show that interaction with the environment quenches the 
inversion transition for what might be described as ‘classical’ reasons.  The 
broadening, shift and quenching of the Rabi oscillation are simply consequences of 
impacts and may be described within the framework of an oscillator subject to white 
noise from the environment.  There is no evidence for localisation onto spatial 
eigenstates.   
Background:⎯At low pressures in the gas phase, the transition between the energy 
eigenstates is observed near 24GHz (0.8 cm–1) in NH3. 1 In ND3 2 the transition is near 
1.6 GHz (0.053 cm–1).  In NH3, broadening is observed at pressures above a few mm 
of mercury, with a shift to lower frequency, and quenching is complete at about 1.7 
bar. In ND3, pressures about 15 times lower yield the same effects, in proportion to 
the inversion transition frequency.     
The first explanation of the shift and broadening of the ammonia inversion 
transition frequency was given by Anderson5 in terms of perturbation by the electric 
dipole-dipole interaction between ammonia molecules.  Anderson’s discussion was 
only qualitative, and Margenau investigated the quantum states of two ammonia 
molecules coupled by their dipole-dipole interaction in more detail.6 He showed that 
the interaction leads to the splitting of the transition into a higher frequency 
component with reduced strength and a lower frequency with increased strength. 
While this accounts for the initial shift to lower frequency, it fails to account for the 
quenching of the inversion transition at a higher pressure.  More recently, the dipole-
dipole interaction model has been treated by a quantum mean-field approximation 
yielding, apparently, a frequency shift, quenching and spatial localisation at pressures 
for NH3 and ND3 in good agreement with experiment.7 
 The standard theory of line-broadening by impact is given by Van Vleck and 
Weisskopf.8  It predicts a line-shape function  
( ) ( )202
−− bb
f  (1) 
where the width b is given by 1/2πτ for strong impacts occurring at a mean interval of 
τ, and therefore proportional to the pressure.  The theory does not predict any peak 
shift: ν0 is a constant, the natural frequency of the oscillator.  Anderson developed the 
theory further and obtained a shift of ν0 to lower frequency equal to the width b.9  
Fano recast the problem of pressure broadening in the Liouville representation and 
obtained a shift to lower frequency independent of the broadening.10  Ben-Reuven 
used the Fano theory to show that the ammonia spectra can be well-fitted with a 
related expression but with three independent parameters proportional to the pressure. 
Two of them express the effects of elastic collisions on the width and on the 
frequency shift, and the third parameter expresses the effect of inelastic collisions.11  
We are interested in a dynamical theory of the transition and of quenching and 
localisation. It is important to know if the dipole-dipole interaction of ammonia 
molecules is crucial to the quenching, or if it merely influences the collision cross-
section while impacts are sufficient to account for the quenching. Accordingly, we 
have set up a molecular dynamics simulation in which the quantum nature of the 
ammonia molecule is explicitly taken into account.12  Here we show that the model 
accounts for the shift, broadening and quenching of the inversions transition purely in 
terms of perturbation by collision with ambient gas molecules. 
The Ammonia Quantum Molecular Dynamics Model:⎯We model the problem in one 
dimension.  The ammonia molecule is modelled by a double-well potential, with the 
two time-dependent spatial wavefunctions ΨL and ΨR. With a weak coupling between 
the wells the Hamiltonian in the spatial basis is 
  (2) ⎟⎟
Diagonalising, the ground and first excited states of the system are found to be Ψ0 and 
Ψ1 with a frequency splitting of ω1. The general state of the system is a superposition, 
with 
Ψ+Ψ=Ψ
 (3) 
Expanding this in the spatial basis set ΨL and ΨR, we have time-varying coefficients, 
 RL tt Ψβ+Ψα=Ψ )()(  (4) 
so that the amplitude of the wave-function beats, or oscillates between the two wells.  
The squared amplitude 2)(tα  = α*α oscillates at the frequency ω1 and with a beat 
amplitude that depends on the initial values of a and b (from zero for e.g. 
2/1== ba  to a maximum amplitude of unity for e.g. a = 1, b = 0). This oscillation 
is the inversion transition or Rabi oscillation of the molecule.  
 We model impacts, or interactions with the environment, by a term which is 
diagonal in the spatial representation. That is, we suppose that the double well is tilted 
during an impact. If a gas atom coming in from the left raises the energy of the left-
hand well, the Hamiltonian during impact is 
  (5) ⎟⎟
Diagonalising and expanding in the spatial basis set as before, we obtain the 
normalised eigenvectors u and v of H′ . Equations 3 and 4 become  
 RLPP ttba Ψβ′+Ψα′=Ψ′+Ψ′=Ψ′ )()(10  (6) 
The Rabi oscillation is now at a much higher frequency and a much smaller amplitude 
(for  ωP >> ω1).  In reality, the perturbation rises and falls continuously in an impact, 
but we approximate with a top-hat function, so that ωP is switched on at a time t0 and 
switched off again at t1. At these times, we match the coefficients in the spatial basis, 
using  and )()( 00 tt α=α′ )()( 00 tt β=β′  to solve for aP and bP at the onset of the 
perturbation, and then the new )()( 11 tt α′=α , )()( 11 tt β′=β  to solve for the new a and 
b at the end of the perturbation. These boundary conditions ensure that the amplitude 
and phase of the wave-function in each well do not change discontinuously at the 
beginning and end of the perturbation. The resulting time evolution of α*α is 
illustrated in Fig.1.  
0 1 2
Time, t
xirta
Fig.1.  The evolution of the occupancy of the left-hand well is shown 
with two perturbations occurring at t = 0.7 and t = 1.6. The Rabi 
angular frequency ω1 is 2π and the perturbation ωP = 60. The initial 
wavefunction is given by a = b = 1/ 2 ; after the two perturbations the 
values are a = 0.54 – 0.73i, b = 0.36 + 0.22i.    
To model NH3 and ND3, we can choose the units of time so that the Rabi 
frequency is unity (ω1 = 2π). The strength of the perturbation is of the order of kBT, 
which at room temperature is 208 cm–1. For NH3, therefore, we take ωP = 208 ω1/0.8  
= 260 ω1 and for ND3, ωP = 208 ω1/0.0.053 = 3925 ω1.  The duration Δt = t0 – t1 of an 
impact is hard to estimate. However, inspection of Fig.1 shows that to achieve a 
strong impact (in the sense of Van Vleck and Weisskopf 8), we need something of the 
order of one cycle of the perturbed Rabi oscillation, i.e. ωPΔt ~2π, while larger values 
will have no extra effect.  We therefore take values of Δt from a random distribution 
over the range 0 to 2π / ωP.  The average frequency of impacts corresponds to the gas 
pressure.  We require an impact cross-section to relate the frequency of impacts to the 
gas pressure quantitatively. Bleaney and Loubser and other authors obtain impact 
cross-sections from the pressure-broadening of the transition, assuming strong 
impacts and using b = 1/2πτ.  We shall see below that such estimates are unreliable, 
and therefore in our simulation we use the measure p impacts per Rabi cycle instead 
of pressure, and we vary p over a wide range.    
We calculate the values of α*α at discrete time intervals δt with Δt < δt << 1.  
At each time interval we have a probability δt/τ of having an impact, so that there are 
p = 1/τ impacts per cycle. If there is an impact, we use )()( 00 tt α=α′  and 
 to solve for aP and bP at the onset of the perturbation, and then 
calculated the new , 
)()( 00 tt β=β′
)()( 11 tt α′=α )()( 11 tt β′=β  to solve for the new a and b at the 
end of the perturbation. Then the calculation of the list of values is resumed. 
Examples are shown in Fig.2 for medium (a) and high (b) values of p. The numerical 
Fourier Transforms of the lists are calculated, shown in Fig.2(c) and (d), and fitted 
with Af(ν) of eq.1, with b, ν0 and amplitude A as fitting parameters.  Our interest here 
is the fitted values of b and ν0 as functions of p.  In Figure 3 these are compared with 
the experimental data for NH3,16 with the constant of proportionality between p and 
pressure (corresponding to the impact cross-section) as a free parameter.  
Fig.2.  The occupation of the left-hand well, y(t) = α*α, is plotted 
against time, for (a) p = 3.5 impacts per cycle, below the quenching, 
and (b) p =  7.5 impacts per cycle, above the quenching.  The Fourier 
transforms Y(ν) are shown in (c) and (d) respectively, together with the 
fits using eq.1.   
Fig.3.  The solid curves show (a) the peak frequency and (b) the 
broadening for NH3 as a function of the number of impacts per cycle 
as described in the text. The data points show (a) the peak frequency 
and (b) the broadening reported by Bleaney and Loubser,16 scaled as 
described in the text.  
The NH3 data is plotted with p = 4.5 impacts per cycle equivalent to a pressure 
P = 1 bar. For ND3, the data fits equally well but with p = 4.5 equivalent to the 
pressure P = 1/15 bar, consistent with the fifteen times lower inversion frequency in 
ND3 but the same impact parameter.  In both cases, full quenching is observed at 
about 6.5 impacts per cycle. The model presented here accounts remarkably well for 
the shift and quenching of the ammonia inversion transition peak. It accounts less well 
for the broadening, which occurs initially at the rate b ~ 0.25 p in the simulation and 
0.18 p in the experimental data. The experimental broadening is about three-quarters 
of the model broadening up to the quenching pressure, and above the quenching 
pressure the experimental broadening continues to increase while the model 
broadening decreases.   To gain a better understanding of this behaviour, we 
investigate how a simple classical oscillator behaves under similar perturbations.  
A Classical Perturbed Harmonic Oscillator:⎯A classical oscillator may be perturbed 
by collision in a large variety of well-defined ways. We evaluate two perturbations 
here. We calculate the values of a sinusoid of frequency ν = 1 at discrete time 
intervals δt << 1.  At each time interval we have a probability δt/τ of having an 
impact, so that there are p = 1/τ impacts per cycle. If there is an impact, the sinusoid is 
modified accordingly, and then the calculation of the list of values is resumed. The 
numerical Fourier Transform is calculated and we find that af(ν) of eq.1 fits well for a 
variety of definitions of the impacts, over a very wide range of p, with b, ν0 and a as 
fitting parameters.  Our interest here is the fitted values of b and ν0 as functions of p.   
Impactsper cycle
ycneuqerF
0 1 2 3 4 5 6
Impactsper cycle
0 1 2 3 4 5 6
Impacts per cycle
ycneuqerF
0 2 4 6 8 10 12 14
Impacts per cycle
0 2 4 6 8 10 12 14
Fig.4.  The peak frequency and the peak width are plotted for the broken 
sinusoids with unity frequency described in the text.  In (a) and (b), results 
are for the strongest possible impact, with phase and amplitude completely 
randomised by the impact.  In (c) and (d), the boundary condition at 
impact keeps the sinusoid continuous but changes the phase and amplitude 
at random within that constraint.   
 The strongest perturbation possible (in the sense of Van Vleck and Weisskopf 
8) is a collision that destroys all memory of position and speed (or amplitude and 
phase).  To model this, at impact we pick the new amplitude A of the sinusoid 
Acos(2πνt + ϕ) at random from the range 0 to 1 and the new phase ϕ at random from 
the range 0 to 2π. In this model, the peak is shifted to higher frequency, shown in 
Fig.4(a) until the width of the transition in Fig.4(b) reaches b = 1, at which point 
quenching occurs, i.e. the frequency collapses to zero.  The width continues to 
increase at still higher values of p, Fig.4(b).  This is a stronger impact than the 
impacts of the ammonia molecule, for the quenching occurs at p = 3.5 impacts per 
cycle and the initial slope of the broadening is given by b ~ 0.5 p.  It is interesting that 
Van Vleck and Weisskopf  give the broadening for strong impacts as 1/2πτ, 
equivalent to 0.16 p. 
 In an alternative definition of impact which is in closer accordance with the 
ammonia model of Section 2.1 and Fig.1, we define the impact at t0 by taking the 
position x(t0) as unchanged by the impact, the new amplitude A as random in the 
range x(t0) to 1, and the new phase ϕ such that the speed is a random variable in )( 0tx&
the range consistent with the new amplitude. In this model, the peak shift in Fig.4(c) 
and width behave in very much the same way as the ammonia results of Fig.3, with 
the initial broadening b ~0.2 p.  However, the impact is weaker than in the ammonia 
model, for quenching occurs at p = 10.5 impacts per cycle. Above quenching b 
decreases again.  
In these models we may weaken the strength of the impact by letting the new 
amplitude and phase be given by some amount ε of the amplitude and phase 
calculated as above plus (1 – ε) of the old amplitude and phase.  Not surprisingly, the 
shift and broadening are identical if plotted as functions of the normalised impact rate 
εp.  
Results and Discussion:⎯Fig.4 shows that the details of the peak shift and the 
broadening are very sensitive to the exact nature of the boundary conditions between 
the periods of unperturbed free oscillation. A more complete description of the impact 
(including, for example, inelastic collisions as in Ben-Reuven11) might well account 
for the discrepancies between data and model in Fig.3. However, we do not know of 
any way to predict the initial slope of b(p), nor its functional form below and above 
the quenching, from a specification of the boundary conditions. Neither the 
mathematics of the noisy oscillator (see, e.g. the book by Gittenberg13) nor of the 
classical kicked rotor (see, e.g. Ref.) appear to answer this question.   
The key result is that the ammonia model (Fig.3) and even the broken sinusoid 
of Fig.4(c) both show the Rabi oscillation frequency shifting to lower frequency, 
broadening, and quenching – going to zero frequency – as the impact rate is increased, 
in agreement with experiment. Note that the density matrix shows no evidence of 
localisation.  In particular, the off-diagonal elements do not go towards zero as 
predicted by the decoherence programme. Nor is there any evidence or quantum state 
diffusion towards the configurational eigenstates. The ammonia shift and quenching 
are fully accounted for in terms of a perturbed oscillator, and should not therefore be 
cited as an experimental observation of quantum localisation.  
Acknowledgements:⎯We are grateful to Prof. I.C. Percival and Dr T. Prellberg for 
useful discussions.    
References  
1. C.H. Townes, Phys. Rev. 70, 665 (1946). 
2. G. Birnbaum and A.A. Maryott, Phys. Rev. 92, 270 (1953).  
3. R.G. Woolley, Adv. Phys. 25, 27 (1976). 
4. E. Joos, in Decoherence and the Appearance of a Classical World in Quantum 
Theory, ed. D. Guilini (Springer, Berlin, 1996).  
5. P.W. Anderson, Phys. Rev. 75, 1450 (1949). 
6. H. Margenau, Phys. Rev. 76, 1423 (1949). 
7. G. Jona-Lasinio, C. Presilla and C. Toninelli, Phys. Rev. Lett. 88, 123001 (2002). 
8. J.H. Van Vleck and V.F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945).  
9. P.W. Anderson, Phys. Rev. 76, 647 (1949). 
10. U. Fano, Phys. Rev. 131, 259 (1963). 
11. A. Ben-Reuven, Phys. Rev. Lett. 14, 349 (1965). 
12. D.J. Dunstan, archive.  
13. M. Gittenberg, The Noisy Oscillator: The First Hundred Years, From Einstein 
Until Now (World Scientific, Singapore, 2005). 
14. B. Bleaney and R.P. Penrose, Proc. Roy. Soc. A189, 358 (1947). 
15. B. Bleaney and R.P. Penrose, Proc. Phys. Soc. 60, 540 (1948). 
16. B. Bleaney and J.H.N. Loubser, Proc. Phys. Soc. A63, 483 (1950). 
	Acknowledgements:(We are grateful to Prof. I.C. Percival and Dr T. Prellberg for useful discussions.

  The mechanism of the shift, broadening and quenching of the ammonia inversion
frequency with gas pressure has been a problem of lively interest for over
seventy years. A simple quantum model of the ammonia molecule perturbed by
collisions with ideal gas molecules displays the essential features of the
experimental data for NH3 and for ND3. The model does not display the behaviour
expected from theories of quantum localisation such as quantum state diffusion
and decoherence. On the other hand, models of perturbed classical oscillators
do display similar behaviour to our model. The quenching of the ammonia
inversion transition cannot therefore be interpreted as spatial localisation of
the wavefunction.

Introduction:⎯Since the early days of microwave spectroscopy, the inversion 
transition of the ammonia molecule NH3 and ND3 has been extensively studied 
experimentally and theoretically. Much of the interest lies in the fact that it is the 
smallest and simplest of the pyramidal and enantiomorphic molecules whose ground 
and excited energy eigenstates are the quantum superpositions of two different spatial 
configurations, and that it is light enough that the transitions between the energy 
eigenstates are fast enough to be experimentally accessible. The ammonia molecule 
has two spatial eigenstates L  and R  with the nitrogen atom on one side or the 
other of the plane of hydrogen atoms, and its energy ground and first excited states 
0  and 1  are the symmetric and antisymmetric quantum superpositions of the 
spatial eigenstates (ignoring rotational and vibrational states).   
The ammonia maser is based upon the transition between the energy 
eigenstates, which may also be described as the Rabi oscillation between the spatial 
eigenstates.  However, the inversion transition is seen only at low gas pressure. As the 
gas pressure is increased, the transition broadens, shifts to lower frequency and then 
quenches (the frequency goes to zero).  The ammonia molecule appears to undergo 
spatial localisation as a result of interaction with the environment. This would be of 
immense theoretical interest. In chemistry and in the classical world generally, 
enantiomorphic molecules with distinguishable spatial eigenstates L  and R  are 
always found in their spatial eigenstates (classical behaviour) rather than their energy 
ground states (quantum behaviour).3  Whilst ammonia is not enantiomorphic, it does 
appear to show both behaviours, quantum at low pressure and classical at high 
pressure, if the quenching is considered to be a direct observation of localisation or 
collapse of the wave-function into a spatial eigenstate.  Within the context of the 
decoherence programme, it has been treated quantitatively in that way.4   
In this paper, we show that interaction with the environment quenches the 
inversion transition for what might be described as ‘classical’ reasons.  The 
broadening, shift and quenching of the Rabi oscillation are simply consequences of 
impacts and may be described within the framework of an oscillator subject to white 
noise from the environment.  There is no evidence for localisation onto spatial 
eigenstates.   
Background:⎯At low pressures in the gas phase, the transition between the energy 
eigenstates is observed near 24GHz (0.8 cm–1) in NH3. 1 In ND3 2 the transition is near 
1.6 GHz (0.053 cm–1).  In NH3, broadening is observed at pressures above a few mm 
of mercury, with a shift to lower frequency, and quenching is complete at about 1.7 
bar. In ND3, pressures about 15 times lower yield the same effects, in proportion to 
the inversion transition frequency.     
The first explanation of the shift and broadening of the ammonia inversion 
transition frequency was given by Anderson5 in terms of perturbation by the electric 
dipole-dipole interaction between ammonia molecules.  Anderson’s discussion was 
only qualitative, and Margenau investigated the quantum states of two ammonia 
molecules coupled by their dipole-dipole interaction in more detail.6 He showed that 
the interaction leads to the splitting of the transition into a higher frequency 
component with reduced strength and a lower frequency with increased strength. 
While this accounts for the initial shift to lower frequency, it fails to account for the 
quenching of the inversion transition at a higher pressure.  More recently, the dipole-
dipole interaction model has been treated by a quantum mean-field approximation 
yielding, apparently, a frequency shift, quenching and spatial localisation at pressures 
for NH3 and ND3 in good agreement with experiment.7 
 The standard theory of line-broadening by impact is given by Van Vleck and 
Weisskopf.8  It predicts a line-shape function  
( ) ( )202
−− bb
f  (1) 
where the width b is given by 1/2πτ for strong impacts occurring at a mean interval of 
τ, and therefore proportional to the pressure.  The theory does not predict any peak 
shift: ν0 is a constant, the natural frequency of the oscillator.  Anderson developed the 
theory further and obtained a shift of ν0 to lower frequency equal to the width b.9  
Fano recast the problem of pressure broadening in the Liouville representation and 
obtained a shift to lower frequency independent of the broadening.10  Ben-Reuven 
used the Fano theory to show that the ammonia spectra can be well-fitted with a 
related expression but with three independent parameters proportional to the pressure. 
Two of them express the effects of elastic collisions on the width and on the 
frequency shift, and the third parameter expresses the effect of inelastic collisions.11  
We are interested in a dynamical theory of the transition and of quenching and 
localisation. It is important to know if the dipole-dipole interaction of ammonia 
molecules is crucial to the quenching, or if it merely influences the collision cross-
section while impacts are sufficient to account for the quenching. Accordingly, we 
have set up a molecular dynamics simulation in which the quantum nature of the 
ammonia molecule is explicitly taken into account.12  Here we show that the model 
accounts for the shift, broadening and quenching of the inversions transition purely in 
terms of perturbation by collision with ambient gas molecules. 
The Ammonia Quantum Molecular Dynamics Model:⎯We model the problem in one 
dimension.  The ammonia molecule is modelled by a double-well potential, with the 
two time-dependent spatial wavefunctions ΨL and ΨR. With a weak coupling between 
the wells the Hamiltonian in the spatial basis is 
  (2) ⎟⎟
Diagonalising, the ground and first excited states of the system are found to be Ψ0 and 
Ψ1 with a frequency splitting of ω1. The general state of the system is a superposition, 
with 
Ψ+Ψ=Ψ
 (3) 
Expanding this in the spatial basis set ΨL and ΨR, we have time-varying coefficients, 
 RL tt Ψβ+Ψα=Ψ )()(  (4) 
so that the amplitude of the wave-function beats, or oscillates between the two wells.  
The squared amplitude 2)(tα  = α*α oscillates at the frequency ω1 and with a beat 
amplitude that depends on the initial values of a and b (from zero for e.g. 
2/1== ba  to a maximum amplitude of unity for e.g. a = 1, b = 0). This oscillation 
is the inversion transition or Rabi oscillation of the molecule.  
 We model impacts, or interactions with the environment, by a term which is 
diagonal in the spatial representation. That is, we suppose that the double well is tilted 
during an impact. If a gas atom coming in from the left raises the energy of the left-
hand well, the Hamiltonian during impact is 
  (5) ⎟⎟
Diagonalising and expanding in the spatial basis set as before, we obtain the 
normalised eigenvectors u and v of H′ . Equations 3 and 4 become  
 RLPP ttba Ψβ′+Ψα′=Ψ′+Ψ′=Ψ′ )()(10  (6) 
The Rabi oscillation is now at a much higher frequency and a much smaller amplitude 
(for  ωP >> ω1).  In reality, the perturbation rises and falls continuously in an impact, 
but we approximate with a top-hat function, so that ωP is switched on at a time t0 and 
switched off again at t1. At these times, we match the coefficients in the spatial basis, 
using  and )()( 00 tt α=α′ )()( 00 tt β=β′  to solve for aP and bP at the onset of the 
perturbation, and then the new )()( 11 tt α′=α , )()( 11 tt β′=β  to solve for the new a and 
b at the end of the perturbation. These boundary conditions ensure that the amplitude 
and phase of the wave-function in each well do not change discontinuously at the 
beginning and end of the perturbation. The resulting time evolution of α*α is 
illustrated in Fig.1.  
0 1 2
Time, t
xirta
Fig.1.  The evolution of the occupancy of the left-hand well is shown 
with two perturbations occurring at t = 0.7 and t = 1.6. The Rabi 
angular frequency ω1 is 2π and the perturbation ωP = 60. The initial 
wavefunction is given by a = b = 1/ 2 ; after the two perturbations the 
values are a = 0.54 – 0.73i, b = 0.36 + 0.22i.    
To model NH3 and ND3, we can choose the units of time so that the Rabi 
frequency is unity (ω1 = 2π). The strength of the perturbation is of the order of kBT, 
which at room temperature is 208 cm–1. For NH3, therefore, we take ωP = 208 ω1/0.8  
= 260 ω1 and for ND3, ωP = 208 ω1/0.0.053 = 3925 ω1.  The duration Δt = t0 – t1 of an 
impact is hard to estimate. However, inspection of Fig.1 shows that to achieve a 
strong impact (in the sense of Van Vleck and Weisskopf 8), we need something of the 
order of one cycle of the perturbed Rabi oscillation, i.e. ωPΔt ~2π, while larger values 
will have no extra effect.  We therefore take values of Δt from a random distribution 
over the range 0 to 2π / ωP.  The average frequency of impacts corresponds to the gas 
pressure.  We require an impact cross-section to relate the frequency of impacts to the 
gas pressure quantitatively. Bleaney and Loubser and other authors obtain impact 
cross-sections from the pressure-broadening of the transition, assuming strong 
impacts and using b = 1/2πτ.  We shall see below that such estimates are unreliable, 
and therefore in our simulation we use the measure p impacts per Rabi cycle instead 
of pressure, and we vary p over a wide range.    
We calculate the values of α*α at discrete time intervals δt with Δt < δt << 1.  
At each time interval we have a probability δt/τ of having an impact, so that there are 
p = 1/τ impacts per cycle. If there is an impact, we use )()( 00 tt α=α′  and 
 to solve for aP and bP at the onset of the perturbation, and then 
calculated the new , 
)()( 00 tt β=β′
)()( 11 tt α′=α )()( 11 tt β′=β  to solve for the new a and b at the 
end of the perturbation. Then the calculation of the list of values is resumed. 
Examples are shown in Fig.2 for medium (a) and high (b) values of p. The numerical 
Fourier Transforms of the lists are calculated, shown in Fig.2(c) and (d), and fitted 
with Af(ν) of eq.1, with b, ν0 and amplitude A as fitting parameters.  Our interest here 
is the fitted values of b and ν0 as functions of p.  In Figure 3 these are compared with 
the experimental data for NH3,16 with the constant of proportionality between p and 
pressure (corresponding to the impact cross-section) as a free parameter.  
Fig.2.  The occupation of the left-hand well, y(t) = α*α, is plotted 
against time, for (a) p = 3.5 impacts per cycle, below the quenching, 
and (b) p =  7.5 impacts per cycle, above the quenching.  The Fourier 
transforms Y(ν) are shown in (c) and (d) respectively, together with the 
fits using eq.1.   
Fig.3.  The solid curves show (a) the peak frequency and (b) the 
broadening for NH3 as a function of the number of impacts per cycle 
as described in the text. The data points show (a) the peak frequency 
and (b) the broadening reported by Bleaney and Loubser,16 scaled as 
described in the text.  
The NH3 data is plotted with p = 4.5 impacts per cycle equivalent to a pressure 
P = 1 bar. For ND3, the data fits equally well but with p = 4.5 equivalent to the 
pressure P = 1/15 bar, consistent with the fifteen times lower inversion frequency in 
ND3 but the same impact parameter.  In both cases, full quenching is observed at 
about 6.5 impacts per cycle. The model presented here accounts remarkably well for 
the shift and quenching of the ammonia inversion transition peak. It accounts less well 
for the broadening, which occurs initially at the rate b ~ 0.25 p in the simulation and 
0.18 p in the experimental data. The experimental broadening is about three-quarters 
of the model broadening up to the quenching pressure, and above the quenching 
pressure the experimental broadening continues to increase while the model 
broadening decreases.   To gain a better understanding of this behaviour, we 
investigate how a simple classical oscillator behaves under similar perturbations.  
A Classical Perturbed Harmonic Oscillator:⎯A classical oscillator may be perturbed 
by collision in a large variety of well-defined ways. We evaluate two perturbations 
here. We calculate the values of a sinusoid of frequency ν = 1 at discrete time 
intervals δt << 1.  At each time interval we have a probability δt/τ of having an 
impact, so that there are p = 1/τ impacts per cycle. If there is an impact, the sinusoid is 
modified accordingly, and then the calculation of the list of values is resumed. The 
numerical Fourier Transform is calculated and we find that af(ν) of eq.1 fits well for a 
variety of definitions of the impacts, over a very wide range of p, with b, ν0 and a as 
fitting parameters.  Our interest here is the fitted values of b and ν0 as functions of p.   
Impactsper cycle
ycneuqerF
0 1 2 3 4 5 6
Impactsper cycle
0 1 2 3 4 5 6
Impacts per cycle
ycneuqerF
0 2 4 6 8 10 12 14
Impacts per cycle
0 2 4 6 8 10 12 14
Fig.4.  The peak frequency and the peak width are plotted for the broken 
sinusoids with unity frequency described in the text.  In (a) and (b), results 
are for the strongest possible impact, with phase and amplitude completely 
randomised by the impact.  In (c) and (d), the boundary condition at 
impact keeps the sinusoid continuous but changes the phase and amplitude 
at random within that constraint.   
 The strongest perturbation possible (in the sense of Van Vleck and Weisskopf 
8) is a collision that destroys all memory of position and speed (or amplitude and 
phase).  To model this, at impact we pick the new amplitude A of the sinusoid 
Acos(2πνt + ϕ) at random from the range 0 to 1 and the new phase ϕ at random from 
the range 0 to 2π. In this model, the peak is shifted to higher frequency, shown in 
Fig.4(a) until the width of the transition in Fig.4(b) reaches b = 1, at which point 
quenching occurs, i.e. the frequency collapses to zero.  The width continues to 
increase at still higher values of p, Fig.4(b).  This is a stronger impact than the 
impacts of the ammonia molecule, for the quenching occurs at p = 3.5 impacts per 
cycle and the initial slope of the broadening is given by b ~ 0.5 p.  It is interesting that 
Van Vleck and Weisskopf  give the broadening for strong impacts as 1/2πτ, 
equivalent to 0.16 p. 
 In an alternative definition of impact which is in closer accordance with the 
ammonia model of Section 2.1 and Fig.1, we define the impact at t0 by taking the 
position x(t0) as unchanged by the impact, the new amplitude A as random in the 
range x(t0) to 1, and the new phase ϕ such that the speed is a random variable in )( 0tx&
the range consistent with the new amplitude. In this model, the peak shift in Fig.4(c) 
and width behave in very much the same way as the ammonia results of Fig.3, with 
the initial broadening b ~0.2 p.  However, the impact is weaker than in the ammonia 
model, for quenching occurs at p = 10.5 impacts per cycle. Above quenching b 
decreases again.  
In these models we may weaken the strength of the impact by letting the new 
amplitude and phase be given by some amount ε of the amplitude and phase 
calculated as above plus (1 – ε) of the old amplitude and phase.  Not surprisingly, the 
shift and broadening are identical if plotted as functions of the normalised impact rate 
εp.  
Results and Discussion:⎯Fig.4 shows that the details of the peak shift and the 
broadening are very sensitive to the exact nature of the boundary conditions between 
the periods of unperturbed free oscillation. A more complete description of the impact 
(including, for example, inelastic collisions as in Ben-Reuven11) might well account 
for the discrepancies between data and model in Fig.3. However, we do not know of 
any way to predict the initial slope of b(p), nor its functional form below and above 
the quenching, from a specification of the boundary conditions. Neither the 
mathematics of the noisy oscillator (see, e.g. the book by Gittenberg13) nor of the 
classical kicked rotor (see, e.g. Ref.) appear to answer this question.   
The key result is that the ammonia model (Fig.3) and even the broken sinusoid 
of Fig.4(c) both show the Rabi oscillation frequency shifting to lower frequency, 
broadening, and quenching – going to zero frequency – as the impact rate is increased, 
in agreement with experiment. Note that the density matrix shows no evidence of 
localisation.  In particular, the off-diagonal elements do not go towards zero as 
predicted by the decoherence programme. Nor is there any evidence or quantum state 
diffusion towards the configurational eigenstates. The ammonia shift and quenching 
are fully accounted for in terms of a perturbed oscillator, and should not therefore be 
cited as an experimental observation of quantum localisation.  
Acknowledgements:⎯We are grateful to Prof. I.C. Percival and Dr T. Prellberg for 
useful discussions.    
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	Acknowledgements:(We are grateful to Prof. I.C. Percival and Dr T. Prellberg for useful discussions.
