THE JOURNAL OF CHEMICAL PHYSICS 132, 134506 共2010兲
Multitime response functions and nonlinear spectra for model quantum
dissipative systems
Mohammad M. Sahrapour1 and Nancy Makri1,2,a兲
1
Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
Department of Chemistry, University of Illinois, Urbana, Illinois 61801, USA
2
共Received 12 June 2009; accepted 8 February 2010; published online 2 April 2010兲
Using iterative evaluation of the real-time path integral expression, we calculate four-time
correlation functions for one-dimensional systems coupled to model dissipative environments. We
use these correlation functions to calculate response functions relevant to third order infrared or
seventh order Raman experiments for harmonic, Morse, and quadratic-quartic potentials interacting
with harmonic and two-level-system dissipative baths. Our calculations reveal the role of potential
features 共anharmonicity and eigenvalue spectrum兲, both on short and long time scales, on the
response function. Further, thermal excitation causes dramatic changes in the appearance of the
response function, introducing symmetry with respect to the main diagonal. Finally, coupling to
harmonic dissipative baths leads to decay of the response function 共primarily along the ␶3 direction兲
and a broadening of the peaks in its Fourier transform. At high temperatures two-level-system baths
are less efficient in destroying coherence than harmonic baths of similar parameters. © 2010
American Institute of Physics. 关doi:10.1063/1.3336463兴
I. INTRODUCTION
The interpretation of vibrational line shapes in liquids
has been the subject of numerous experimental and theoretical investigations. In recent years, nonlinear optical techniques, which measure the response of liquids subjected to a
sequence of laser pulses, have been added to the more traditional linear spectroscopic studies. Nonlinear experiments
serve as sensitive and highly informative probes of the dynamics that can distinguish between homogeneous and inhomogeneous contributions.1–6
Nonlinear spectroscopic signals are, by their nature,
more complex than their linear counterparts, and contain rich
information. In particular, nonlinear spectroscopy provides a
superb way of probing the Hamiltonian of a molecular system, as well as its coupling to the environment. The main
advantage of these experiments is that different order spectroscopies are sensitive to anharmonic terms through different orders, thus in principle, enabling one to reconstruct the
molecular potential. Deciphering these features rests largely
on theoretical calculations. Even though the theoretical
framework is available,7 accurate calculation of nonlinear vibrational spectroscopic signals in polyatomic systems is impractical. Thus, many important insights have come out of
model studies. Tanimura and Mukamel8 obtained analytical
results for a harmonic oscillator coupled to a harmonic dissipative bath. Cao and co-workers9,10 calculated microcanonical response functions for a Morse oscillator and analyzed the quantum-classical correspondence. Loring and
co-workers11–15 studied anharmonic oscillator models using
classical and semiclassical approximations. In a series of recent papers, Tanimura and co-workers16–21 presented numerical calculations of response functions for harmonic and ana兲
Electronic mail: [email protected].
0021-9606/2010/132共13兲/134506/12/$30.00
harmonic oscillators interacting with Drude-type dissipative
harmonic baths using a high-temperature Fokker–Planck
treatment and its low-temperature extension.
As is well known, the simulation of the quantum dynamics of condensed phase processes remains a challenging
problem. In the special case of a low-dimensional 共or fewlevel兲 system in contact with a dissipative harmonic bath,
path integral methods offer an excellent starting point. Feynman and Vernon showed that the harmonic bath can be integrated out exactly within the path integral framework. However, this process introduces terms that are nonlocal in time,
resembling the memory terms in the generalized Langevin
equation. This nonlocality in time prevents evaluation of the
path integral by stepwise methods commonly employed for
wave function propagation. Nevertheless, earlier work in our
group has shown that the path integral can be evaluated iteratively by multiplying a multitime reduced density matrix,
which spans the length of the bath-induced memory, by an
appropriate propagator.22–28 Once converged, this procedure
leads to numerically exact results over very long propagation
times.
In this paper we present converged, fully quantum mechanical path integral calculations of the response function
corresponding to seventh order off-resonant Raman spectroscopy 共or, equivalently, third order resonant infrared spectroscopy兲 for model one-dimensional systems coupled to a dissipative harmonic bath characterized by a spectral density of
the Ohmic form. We also present the two-dimensional frequency spectra for these systems by Fourier transforming the
response functions. The results of our simulations are accurate over hundreds of oscillation periods, offering insights
into subtle spectroscopic signatures of potential symmetry
and anharmonicity, as well as the dissipative role of the environment, over a wide temperature range.
132, 134506-1
© 2010 American Institute of Physics
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-2
J. Chem. Phys. 132, 134506 共2010兲
M. M. Sahrapour and N. Makri
In Sec. II we describe the theoretical formalism and
summarize the numerical path integral methodology that we
employ. In Secs. III and IV we show the results of our simulations and discuss their prominent features. Finally, in Sec.
V we summarize our results and present some concluding
remarks.
V̂共t兲 = ␣ˆ E2共t兲,
where ␣共q兲 is the polarizability. The main difference between these two types of molecule-laser coupling is in the
order of the electric field. Thus nth order infrared spectroscopy is described by a formulation that is similar to that for
共2n + 1兲th order Raman spectroscopy 共the extra “1” comes
form the laser pulse that causes the Raman signal兲.7 Due to
this equivalence, from now on we will refer only to polarizability and Raman spectroscopy, but our results will apply
equally to nonlinear infrared experiments.
As is well known, in the case of a harmonic system
potential, linear polarizability does not give rise to a nonlinear signal, thus our model for the polarizability includes a
quadratic term,
II. THEORETICAL FORMALISM
Throughout this paper we restrict attention to Hamiltonians of the type
Ĥ = H0共p̂q,q̂兲 + HB共p̂,x̂,q̂兲,
共2.1兲
where
Ĥ0 =
p̂2q
+ V0共q̂兲
2m
共2.2兲
␣共q兲 = ␣1q + ␣2q2 .
describes the subsystem of interest 共with coordinate q̂ and
conjugate momentum p̂q兲, and HB is the Hamiltonian that
models the environment and its interaction with the system.
Specific forms of the bath Hamiltonian are considered in
Secs. III and IV.
In the case of infrared experiments, interaction with the
laser field amounts to an additional term in the Hamiltonian,
which is given by
ˆ E共t兲,
V̂共t兲 = ␮
i
ប3
P共t兲 = Tr关␣ˆ ␳ˆ 共t兲兴.
t
t1
dt1
t0
t2
dt2
t0
共2.6兲
The density operator ␳ˆ can be expanded in powers of the
potential V共t兲 in the interaction picture 共where the evolution
due to H is “rotated out”兲. The third order term in this expansion will have three factors of V共t兲 and thus will correspond to a seventh order Raman 共or third order infrared兲
experiment. The contribution to the polarization from this
term can be written as
共2.3兲
冕 冕 冕
共2.5兲
The polarization is defined as
where E共t兲 is the electric field and ␮共q兲 is the dipole moment
function. For Raman experiments the relevant term is quadratic in the electric field,
␳ˆ I共3兲共t兲 = −
共2.4兲
ˆ
ˆ
P共3兲共t兲 = Tr关␣ˆ e−iHt/ប␳ˆ I共3兲eiHt/ប兴,
共2.7兲
7
where the third order term in the density operator is
dt3关V̂I共t1兲,关V̂I共t2兲,关V̂I共t3兲, ␳ˆ I共t0兲兴兴兴
共2.8兲
t0
and
ˆ
ˆ
V̂I共t兲 = eiHt/ប␣ˆ E2共t兲e−iHt/ប .
共2.9兲
We assume that the molecular system and its environment are initially in thermal equilibrium at a temperature T, such that
1
ˆ
␳ˆ I共t0兲 = e−␤H ,
Z
共2.10兲
ˆ
where Z = Tr e−␤H is the partition function and ␤ = 1 / kBT, where kB is Boltzmann’s constant. Substituting this expression for
␳ˆ I共t兲 into the above and simplifying further, the third order polarization can be written as7
P共3兲共t兲 =
冕 冕 冕
⬁
d␶1
0
⬁
0
d␶2
⬁
d␶3E共t − ␶3兲2E共t − ␶3 − ␶2兲2E共t − ␶3 − ␶2 − ␶1兲2R共7兲共␶3, ␶2, ␶1兲.
共2.11兲
0
Here we have introduced the response function R共7兲, which by expanding the commutators in Eq. 共2.8兲 can be shown to be7
R共7兲共␶3, ␶2, ␶1兲 =
2
ˆ 共i/ប兲Ĥ共␶ +␶ +␶ 兲
ˆ
ˆ
ˆ
−␤H
1 2 3 ␣
ˆ e−共i/ប兲H␶3␣ˆ e−共i/ប兲H␶2␣ˆ e−共i/ប兲H␶1␣ˆ 兲
e
3 Im兵Tr共e
ប
ˆ
ˆ
ˆ
ˆ
− Tr共e−␤H␣ˆ e共i/ប兲Ĥ共␶1+␶2+␶3兲␣ˆ e−共i/ប兲H␶3␣ˆ e−共i/ប兲H␶2␣ˆ e−共i/ប兲H␶1兲
ˆ
ˆ
ˆ
+ Tr共e−␤H␣ˆ e共i/ប兲Ĥ␶1␣ˆ e共i/ប兲Ĥ共␶2+␶3兲␣ˆ e−共i/ប兲H␶3␣ˆ e−共i/ប兲H共␶2+␶1兲兲
ˆ
ˆ
ˆ
− Tr共e−␤He共i/ប兲Ĥ␶1␣ˆ e共i/ប兲Ĥ共␶2+␶3兲␣ˆ e−共i/ប兲H␶3␣ˆ e−共i/ប兲H共␶2+␶1兲␣ˆ 兲其.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
共2.12兲
134506-3
J. Chem. Phys. 132, 134506 共2010兲
Response functions and nonlinear spectra
Each of the terms in Eq. 共2.12兲 is a four-time autocorrelation
function of the polarizability operator. In order to compare
our results with previous simulations and experiments, we
set ␶2 to zero. In Sec. III, we calculate R共7兲共␶3 , 0 , ␶1兲
for harmonic and anharmonic systems coupled to model
dissipative environments.
We use the path integral methodology developed earlier
in our group22–24,26–32 to evaluate the response function. With
the choice ␶2 = 0, a simple rearrangement shows that each
term in Eq. 共2.12兲 takes the form
1
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
CABCD共t1,t3兲 = Tr共e−␤HD̂eiHt3/បĈe−iHt3/បB̂eiHt1/បÂe−iHt1/ប兲 = Tr共e−␤HD̂Ĉ共t3兲B̂Â共t1兲兲,
Z
Z
共2.13兲
where Â, B̂, Ĉ, and D̂ are equal to 1, ␣ˆ , or ␣ˆ 2. Using the permutation invariance of the trace once again, we rewrite the
correlation function as27,28
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
CABCD共t1,t3兲 = Tr共e−␤H/2e−iHt1/បD̂eiHt3/បĈe−iHt3/បB̂eiHt1/បÂe−␤H/2兲
Z
=
冕 冕
dqa0
ˆ
ˆ
ˆ
We partition the reciprocal temperature into 2M imaginary
time slices of length ⌬␤ = ␤ / 2M and the two real-time arguments 共assumed multiples of the same time step兲 into N1 and
N3 time steps of lengths ⌬t = t1 / N1 = t3 / N3, respectively. We
employ the quasiadiabatic factorization of the time evolution
operators,29
ˆ
ˆ
ˆ
ˆ
e−iH⌬␶/ប ⬇ e−iHB⌬␶/2បe−iH0⌬␶/បe−iHB⌬␶/2ប
共2.15兲
共where ⌬␶ = ⌬t or −iប⌬␤兲. By diagonalizing the system position operator q̂ in the basis of the n low-lying eigenstates
⌽i of the system Hamiltonian that carry appreciable population at the given temperature, we construct a potentialoptimized discrete variable representation 共DVR兲,33
n
ui = 兺 Lij⌽ j ,
共2.16兲
j=1
such that the discretized position states and eigenvalues satisfy the relation
具ui兩q兩u j典 = qi␦ij .
ˆ
ˆ
ˆ
dqb0具qb0兩TrB共e−␤H/2e−iHt1/បD̂eiHt3/បĈe−iHt3/បB̂eiHt1/បÂe−␤H/2兲兩qa0典␦共qa0 − qb0兲.
共2.17兲
Evaluating the real and imaginary time propagators in this
basis and integrating out the Gaussian bath, the correlation
function takes the form of a path integral34 共discretized on
the DVR grid兲 with respect to the system of interest, where
the effects of the bath enter through the Feynman–Vernon
influence functional35 evaluated at the DVR eigenvalues.30
Full summation of the path integral typically involves an
astronomical number of terms, equal to n2共M+N1+N3兲. At the
same time, Monte Carlo methods fail to converge 共except at
short times兲 due to the oscillatory nature of the multidimensional integrand.36–38 Even though the presence of the influence functional has introduced nonlocal interactions35 prohibiting a step-by-step evaluation, earlier work in our group
has shown22,23,31 that the extent of nonlocality is finite and
usually much shorter than the desirable propagation time.
共2.14兲
This fact, which is a consequence of decoherence induced by
the bath,22,23 as well as thermal fluctuations along the imaginary time part of the integration contour,28 implies that the
path integral can be decomposed into a series of lowdimensional operations. Specifically, rather than attempting
direct summation of all paths spanning the propagation time,
one needs to carry each summation over only those path
segments that span the memory time.22,23,28 Since the number
of terms grows exponentially with the number of time steps,
such a decomposition results in a dramatic reduction in effort
and enables evaluation of the path integral for long times.
In the case of equilibrium correlation functions of the
type considered in this work, the integration must be performed inward from both ends of the imaginary time
contour,27 as shown in Fig. 1. Assuming that nonlocal influence functional interactions drop off within at most ⌬kmax
time steps both in real and imaginary time, the second summation in the exponent of the influence functional is truncated. The procedure begins by multiplying a
array
with
arguments
2⌬kmax-dimensional
a
b
b b
,
q
,
q
,
.
.
.
,
q
by
a
2⌬k
⫻ 2⌬kmax
qa0 , qa1 , . . . , q⌬k
max
0 1
⌬kmax−1
max−1
dimensional propagator matrix that contains all nonzero in-
q0a
−t1
q1a
a
qM
+ 1
a
qM
+ 1 −1
t3
q
b
qM
+ 1
b
qM
∆t
− i β
Reτ
∆β
M
a
qM
+ 1 + 3
b
qM
+ 1 + 3
q0b
FIG. 1. Schematic representation of the time contour followed in the iterative evaluation of the path integral expression for Eq. 共2.14兲.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-4
J. Chem. Phys. 132, 134506 共2010兲
M. M. Sahrapour and N. Makri
fluence functional interactions. The propagated array is multiplied by the propagator matrix again, and the procedure is
repeated along the contour indicated through the arrows in
Fig. 1, including action by the system operators at the designated times. The desired correlation function is obtained by
setting qaM+N +N = qbM+N +N and taking the trace of the propa1
3
1
3
gated array.
The iterative procedure summarized above converges to
the exact quantum mechanical result as the time steps become sufficiently small and the memory lengths chosen to
truncate the influence functional interactions become sufficiently large. Because convergence is easy to test, the calculation leads to numerically exact results that can be used to
obtain insights into the features of nonlinear spectroscopic
signals.
III. RESPONSE FUNCTIONS AND TWO-DIMENSIONAL
SPECTRA FOR MODEL SYSTEMS COUPLED TO
A HARMONIC BATH
A common and very useful model for the environment is
that of a harmonic bath,
冋
ĤB = 兺
i
冉
p̂2i
1
f i共q̂兲
+ mi␻2i x̂i −
2mi 2
mi␻2i
冊册
2
.
共2.18兲
The bath coordinates xi are denoted collectively by the vector
x. True dissipation is achieved when the bath consists of an
infinite number of degrees of freedom. Although the systembath coupling functions f i共q兲 can have an arbitrary form, in
this work we restrict attention to bilinear coupling,
f i共q兲 = ciq.
共2.19兲
The bath frequencies and coupling constants are collectively
specified in terms of the spectral density function39
J共␻兲 = 兺
i
c2i
␦共␻ − ␻i兲.
m i␻ i
共2.20兲
The Hamiltonian of Eq. 共2.18兲 arises naturally in the case of
a crystalline solid, where the potential along small displacement atomic coordinates is quadratic to an excellent approximation, and thus can be brought into the normal mode form.
In that case each bath oscillator represents a particular lattice
vibration or phonon mode. In other situations the harmonic
bath coordinates do not correspond to microscopic modes,
but may represent fictitious degrees of freedom whose collective effect on the system is equivalent to that of the actual
anharmonic environment.34,40–42
The harmonic bath model has been used extensively as a
prototype of dissipation. In the classical limit, the harmonic
bath leads43 to the well-known generalized Langevin equation of motion, where the conventional force exerted on the
system is supplemented by a “random” component as well as
a frictional term. By contrast, the fully quantum mechanical
picture is considerably more complicated and leads to a variety of phenomena governed by the interplay between coherence and dephasing.
In this section we present numerical results for the response function of model one-dimensional systems interacting with a harmonic bath characterized by an Ohmic spectral
density with exponential cutoff,39,44
J共␻兲 = ␥␻e−␻/␻c ,
共2.21兲
where ␥ is the friction constant and ␻c is the bath cutoff
frequency.
The influence functional from a harmonic bath assumes
a convenient analytic form,35 and propagators in the iterative
decomposition of the path integral are given by closed-form
expressions.23 The memory length is treated as a convergence parameter.
A. Harmonic oscillator
Response functions for quadratic Hamiltonians can be
evaluated analytically. In this section we apply the iterative
path integral method to a harmonic oscillator linearly
coupled to a harmonic bath with a linear-quadratic polarizability operator 共␣2 = 0.01␣1兲. Besides verifying the accuracy
of the numerical path integral methodology over very long
simulation intervals, the resulting graphs are useful for comparison, highlighting the consequences of anharmonicity in
the systems that follow. To facilitate a quantitative comparison against the results for the Morse potential presented in
Sec. III B, the mass is m = 115 666.3483 a.u. and the frequency is chosen to match the harmonic frequency of the
Morse oscillator, ␻0 = 214.74 cm−1, which corresponds to a
period 156.8 fs. The temperature used in our calculations is
10 K, corresponding to ប␻␤ = 30.9. In agreement with the
analytical prediction, our numerical results are independent
of temperature. Since the response function vanishes in this
case for a linear polarizability operator,17,45 our calculations
were performed with a linear-quadratic polarizability.
The numerical path integral results for the response
function are shown in Fig. 2. In agreement with available
analytical expressions,17,45 constant-amplitude sinusoidal oscillations with a period of 156.4 fs are observed in the absence of dissipation 关Fig. 2共a兲兴, with a squared oscillatory
pattern observable along the ␶3 direction. The long-time stability and accuracy of our method were verified by carrying
these calculations out to 6 ps in both time directions. Our
path integral results for the harmonic system-bath model are
in good qualitative agreement with the shorter time results
shown in Refs. 19 and 46 for the Drude dissipation. As in the
case of two-time correlation functions,14 coupling to a dissipative environment causes the oscillation amplitude of the
four-time correlation functions to decay in both time directions, but the decay is faster along the ␶3 axis.
Figure 3 shows the absolute value of the Fourier transform of the response function. For the bare one-dimensional
harmonic system, the response function is a product of a
cosine factor in ␶1 and a squared cosine factor in ␶3; as a
result, the Fourier transform is a sum of delta functions products of the type ␦共␻1 ⫾ ␻0兲␦共␻3 ⫾ 2␻0兲 and ␦共␻1 ⫾ ␻0兲␦共␻3兲.
These peaks are seen in Fig. 3共a兲 at ⫾214 cm−1 in the ␻1
direction, and at 0 and ⫾428 cm−1 in the ␻3 direction, but
the expected delta-function lines are hard to observe within
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-5
J. Chem. Phys. 132, 134506 共2010兲
Response functions and nonlinear spectra
Ă Ϳ
500
400
300
200
ω3
100
0
−100
−200
−300
−400
−500
−500
ď Ϳ
−400
−300
−200
−100
−400
−300
−200
−100
0
100
200
300
400
500
0
100
200
300
400
500
ω1
500
400
300
200
ω3
100
0
−100
−200
−300
−400
−500
−500
ω1
FIG. 3. Absolute value of the Fourier transform of the response function the
harmonic models with linear-quadratic polarizability. 共a兲 One-dimensional
harmonic oscillator. 共b兲 Harmonic oscillator coupled to a harmonic bath.
FIG. 2. Response function for harmonic models with linear-quadratic polarizability. 共a兲 One-dimensional harmonic oscillator. 共b兲 Harmonic oscillator
coupled to a harmonic bath.
the resolution of the graph. These ridges become clearly visible in Fig. 3共b兲, where the delta functions are broadened by
virtue of the dissipative interactions.
B. Morse potential
The Morse oscillator offers an important model for vibrational spectroscopy. Nonlinear response functions for
Morse oscillators have been reported in several papers using
quantum mechanical methods and classical or semiclassical
approximations.9–16,21,47 Several of these treatments have focused on calculating the vibrational echo. In addition, solvent effects have been studied by coupling a Morse oscillator
to a dissipative harmonic bath using adiabatic/weak coupling
approximations, the high-temperature quantum Fokker–
Planck equation, and its low-temperature extension.
In this section we present numerically exact results on
the response function for a Morse potential,
V0共q兲 = D0共1 − e−␭q兲2 .
45
共2.22兲
Following earlier studies, we choose the parameters of the
system to mimic those for the iodine molecule, for which the
reduced mass is m = 115 666.3483 a.u., D0 = 12436 cm1, and
␭ = 1.868 Å−1. We performed calculations with linear 共␣2
= 0兲 and linear-quadratic 共␣2 = 0.01␣1兲 polarizability and calculated the response function in each case with and without
linear coupling to the harmonic oscillator bath. The temperature was equal to 10 K, corresponding to ប␻␤ = 30.9. At this
temperature the system is only mildly anharmonic since the
dynamics is governed by motion near the bottom of the potential well.
By contrast to the case of a harmonic potential, where a
quadratic term in the polarizability is necessary to obtain a
nonvanishing response function, for the Morse potential the
addition of the quadratic term does not have such a drastic
effect. While the results of the simulation do change in the
second or third significant figure upon adding the quadratic
term, the latter does not lead to qualitative changes in the
calculated response functions. This behavior, which was observed both with and without coupling to the bath, can be
attributed to the small magnitude of ␣2 in relation to the
linear term. 共As can be seen in the figures, the response function is ⬃106 times smaller in the case of the harmonic oscillator, where the linear term makes no contribution.兲 This is to
be expected again because in the harmonic case the response
function can be shown17,45 to be proportional to ␣21␣22,
whereas in the Morse case it is proportional to ␣41.
Figure 4 shows the response function of the Morse os-
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-6
J. Chem. Phys. 132, 134506 共2010兲
M. M. Sahrapour and N. Makri
−5
x 10
1.5
2000
1800
1
1600
3
τ (fs)
1400
0.5
1200
0
1000
800
−0.5
600
400
−1
200
0
0
500
1000
τ (fs)
1500
2000
−1.5
1
FIG. 5. Response function for the Morse oscillator with linear polarizability
coupled to a harmonic bath.
FIG. 4. Response function for the one-dimensional Morse oscillator with
linear polarizability.
cillator in the absence of dissipation. The response function
for this system appears very different from that obtained with
a harmonic potential. On a short time scale 关Fig. 4共a兲兴 the
response function exhibits oscillations along both time directions. In the ␶1 direction there are constant-amplitude oscillations with a period of 145 fs. In the ␶3 direction one observes oscillations with a period of 157 fs. Unlike the
harmonic case though, the peaks of the response function
alternate in sign along both time directions. Further, the contours are now rotated by 45° and do not change sign along
the ␶1 + ␶2 = 0 direction. Qualitatively similar features were
observed by other groups.9,15,45
Figure 4共b兲, which shows the response function on a
much longer time scale, reveals a sinusoidal modulation of
the response function with a frequency of 17.9 ps. The period
of this modulation, which exceeds the fundamental oscillation period by two orders of magnitude, corresponds to the
difference in spacing of successive energy levels of the
Morse potential and thus is a manifestation of anharmonicity.
This simulation was carried out up to ␶1 = 2.5 ps and ␶3
= 75 ps; based on this region we believe that the above
stated features continue for even longer times. Interestingly,
even though the peaks of the response function lie along
diagonal directions, the observed modulation pattern is parallel to the ␶1 axis.
As seen in Fig. 5 and in agreement with Tanimura’s
calculations,45 the addition of dissipation leads to decay of
the oscillatory pattern. In sharp contrast to the harmonic oscillator, the decay is now much faster 共⬃1 ps兲 along the ␶1
direction, with the oscillations along the ␶3 direction persisting for several picoseconds. This unexpected behavior is a
consequence of the modulation in the ␶3 direction, which was
observed for the Morse system in the absence of dissipation:
In the case of the uncoupled system this modulation leads to
an increase in oscillation amplitude along the ␶3 direction
during the initial few picoseconds, and this increase causes
the effects of dissipation to be less prominent. Eventually,
the quenching in oscillation amplitude induced by the
system-bath coupling becomes dominant, leading to a decay
of the response function along the ␶3 direction after ⬃4 ps.
The two-dimensional spectra for the Morse potential
plots are shown in Fig. 6. In the ␻1 direction there are peaks
at ⫾214 cm−1. There are closely spaced peaks at
⫾212 cm−1 in the ␻3 direction, separated by 1.7 cm−1. This
splitting corresponds to the 18 ps ␻3 modulation seen in the
time domain.
C. Oscillator with quartic anharmonicity
Here we present path integral results for a symmetric,
strongly anharmonic oscillator described by the potential
function
V共x兲 = 21 m␻20x2 + 0.1x4 ,
共2.23兲
with m = 1 and ␻0 = 冑2. This model Hamiltonian is characterized by a large anharmonicity and has been employed as a
model in calculations of linear and nonlinear correlation
functions.15,48–50 Unlike the Morse oscillator, this potential of
Eq. 共2.23兲 is symmetric and its eigenvalue spacing increases
with energy. We present results for six values of the inverse
temperature ␤ corresponding to low 共ប␻0␤ = 30兲, intermediate 共ប␻0␤ = 3, 1.5, and 1兲, and high 共ប␻0␤ = 0.5 and 0.1兲
temperatures.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-7
J. Chem. Phys. 132, 134506 共2010兲
Response functions and nonlinear spectra
300
200
200
100
100
ω3
ω3
300
0
0
−100
−100
−200
−200
−300
−300
−200
−100
0
ω1
100
200
300
−300
−300
−200
−100
0
ω1
100
200
300
FIG. 6. Absolute value of the Fourier transform of the response function for the Morse potential, without 共left兲 and with 共right兲 coupling to a harmonic bath.
The computed response functions for the onedimensional quartic oscillator are shown in Fig. 7. Just as
observed for the Morse potential, the peaks in these figures
lie along diagonal lines; this seems to be a prominent feature
of anharmonicity. At the lowest temperature, we observe
again a modulation of the sinusoidal oscillations along the ␶3
direction, but this modulation now occurs on a shorter time
scale as compared to the Morse potential. This is a consequence of the larger anharmonicity of the quartic oscillator
model. As the temperature is increased, we also observe a
modulation of the oscillations along the ␶1 direction, which
becomes more prominent as the temperature is increased.
A more complex behavior emerges at high temperature.
Even though the contours of the response function still lie
along constant ␶1 + ␶3 lines in the bulk of the graph, we observe a 90° rotation near the main diagonal 共␶1 ⯝ ␶3兲, where
nonalternating sign contours are elongated along and parallel
to the ␶1 = ␶3 line. This is a consequence of very severe
modulation of increasing frequency as the temperature is
raised. In addition, the oscillation frequency is higher at this
temperature, as eigenvalue spacings increase with excitation
energy. We note that the response function is nearly symmetric with respect to the main diagonal and exhibits a ridge
along this diagonal that dominates at high temperatures.
It has been shown10,51,52,47,53,54 that classical nonlinear
response functions of a regular system in thermal equilibrium exhibit a divergence at the echo condition, ␶1 = ␶3. Leegwater and Mukamel52 pointed out that for noninteracting
quartic oscillators at fixed ␶3, this divergence in ␶1 can be
removed by thermal averaging. Indeed, for the linear response function thermal averaging results in a finite classical
response.54 Noid and co-workers53 demonstrated the removal
of the divergence in ␶3 at fixed ␶1 for a thermal ensemble of
Morse oscillators, but also found that a divergence still persists at the echo condition.
Our fully quantum mechanical calculations show no
signs of divergent behavior at long times. Nonetheless, the
height of the ridge observed along the ␶1 ⯝ ␶3 increases logarithmically with increasing temperature. In agreement with
the findings of Loring and co-workers,15,53 at low to moderate temperatures 共ប␻␤ = 30 and 3兲 our response function exhibits long-time oscillations along the line ␶1 = ␶3 at a fre-
quency corresponding to the anharmonicity of the potential.
At higher temperatures we observe recurrences of decreasing
amplitude along this direction, which are eventually
quenched completely at the highest temperature 共ប␻␤ = 0.1兲.
Figure 8 shows the response function with linear coupling to the harmonic bath. Once again, interaction with the
bath leads to an interesting interplay between amplitude
modulation and decay. This is seen very clearly at the lower
temperatures and causes a shift of the peak in the ␶3 direction
to smaller time values. As also observed in the calculations
on harmonic and Morse systems, the decay of the response
function is faster in the ␶3 direction at the lower temperatures. The decay sets in faster as the temperature increases.
Further, the decay rates along the ␶1 and ␶3 directions become comparable at high temperatures, and the response
function again becomes symmetric along the main diagonal.
Fourier transforms of the response functions for this system are shown in Fig. 9. In all cases we clearly see two close
peaks in the ␻3 direction, corresponding to the ␶3 modulations. At the low temperature 共ប␻0␤ = 30兲 only one peak is
seen in the ␻1 direction. Multiple peaks emerge at intermediate temperatures 共ប␻0␤ = 3 and 1.5兲 along both frequency
axes. The presence of these peaks reflects the modulation
observed in the time domain graphs. In all the above cases,
interaction with the bath serves to blur these features, and the
characteristics observed at different temperatures become
less pronounced.
Finally, we also carried out simulations including a cubic
term in the potential at low to intermediate temperatures 共up
to ប␻0␤ = 1.5兲. The resulting response functions were very
similar to the ones shown for the symmetric system with
quartic anharmonicity. Thus it appears that the asymmetry in
the potential plays only a minor role at these temperatures.
IV. RESPONSE FUNCTIONS AND TWO-DIMENSIONAL
SPECTRA FOR AN ANHARMONIC OSCILLATOR
COUPLED TO A TWO-LEVEL-SYSTEM BATH
In this section we consider a nonlinear environment
modeled in terms of a bath of two-level systems 共TLSs兲, as
described by the Hamiltonian
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-8
J. Chem. Phys. 132, 134506 共2010兲
M. M. Sahrapour and N. Makri
FIG. 7. Response function for the one-dimensional quartic oscillator. The axes are in units of the harmonic period. 共a兲 ប␻0␤ = 30. 共b兲 ប␻0␤ = 3. 共c兲 ប␻0␤
= 1.5. 共d兲 ប␻0␤ = 1. 共e兲 ប␻0␤ = 0.5. 共f兲 ប␻0␤ = 0.1.
1
ĤB = − 兺 ប␻i␴ix − q 兺 ci
i
i 2
冑
ប i
␴.
2␻i z
共2.24兲
Here ␴x and ␴z are the usual Pauli spin matrices. The
TLSs have tunneling splittings 2ប␻i, and ci are coupling
coefficients.
As long as the bath corresponds to a truly dissipative
environment, a theoretical analysis has shown26 that the path
integral can still be evaluated by an iterative method; in that
case the relevant propagator does not assume a closed form,
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-9
Response functions and nonlinear spectra
J. Chem. Phys. 132, 134506 共2010兲
FIG. 8. Response function for the quartic oscillator coupled to a harmonic bath. The time axes are in units of the harmonic period. 共a兲 ប␻0␤ = 30. 共b兲 ប␻0␤ = 3.
共c兲 ប␻0␤ = 1.5. 共d兲 ប␻0␤ = 1. 共e兲 ប␻0␤ = 0.5. 共f兲 ប␻0␤ = 0.1. Note the different ranges of the axes in panel 共f兲.
but can be expressed in terms of the influence functional.
In the case where the bath consists of noninteracting degrees
of freedom, the influence functional factorizes and thus
can be evaluated numerically by one-dimensional
procedures.55 However, it is also known that in the
limit where the bath has an infinite number of TLSs, its
effect on the dynamics of the system is equivalent to that
of an effective harmonic bath similar to that in Eqs.
共2.18兲–共2.20兲, whose spectral density is specified by the
relation
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-10
;Ă Ϳ
J. Chem. Phys. 132, 134506 共2010兲
M. M. Sahrapour and N. Makri
2.5
2
2
1.5
1.5
1
1
0.5
0.5
3
ω3
2.5
0
ω
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2.5
−2.5
−2.5
−2.5
−2
−1.5
−1
−0.5
0
ω
0.5
1
1.5
2
2.5
−2
−1.5
−1
−0.5
1
ď Ϳ
1.5
1
1
0.5
0.5
3
3
2
1.5
−1
−1
−1.5
−2
−2
−2
−1.5
−1
−0.5
0
ω1
0.5
1
1.5
2
−2.5
−2.5
2.5
2.5
2
2.5
−2
−1.5
−1
−0.5
ω1
0
0.5
1
1.5
2
2.5
−2
−1.5
−1
−0.5
ω1
0
0.5
1
1.5
2
2.5
−4
−3
−2
−1
0
1
2
3
2.5
2
2
1.5
1.5
1
1
0.5
0.5
ω3
ω3
1.5
−0.5
−1.5
0
FIG. 9. Absolute value of the Fourier
transform of the response function for
the quartic potential without 共left兲 and
with 共right兲 coupling to the bath. 共a兲
ប␻0␤ = 30, 共b兲 ប␻0␤ = 3, 共c兲 ប␻0␤
= 1.5, and 共d兲 ប␻0␤ = 0.1.
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
Ě Ϳ
1
0
ω
ω
0
−2.5
−2.5
0.5
2.5
2
−0.5
Đ Ϳ
0
1
2.5
−2.5
−2.5
ω
−2
−1.5
−1
−0.5
0
ω1
0.5
1
1.5
2
2.5
−2.5
−2.5
5
4
4
3
3
2
2
1
ω3
ω
3
1
0
0
−1
−1
−2
−2
−3
−3
−4
−4
−4
−3
−2
−1
0
ω1
1
2
3
Jeff共␻, ␤兲 = tanh共 21 ប␻␤兲J共␻兲.
4
−5
−5
ω1
共2.25兲
This relation was first established using second order perturbation theory,44 but has also been shown to be exact42 because the effective harmonic bath produces the same influence functional with that of the original TLS bath. This
mapping of a nonlinear medium onto a Gaussian bath holds
rigorously, irrespective of the magnitude of the overall cou-
4
5
pling strength. The above equivalence, whose classical analog is often justified in the spirit of the central limit
theorem,34,40,41 is meaningful only in the context of modulating the dynamics of the observable system. Notice that the
parameters of the equivalent effective bath are, in general,
temperature dependent.
Here we investigate the effects of a nonlinear environment on the response function using the TLS bath model
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-11
J. Chem. Phys. 132, 134506 共2010兲
Response functions and nonlinear spectra
4
30
0.1
3
25
2
0.05
20
0
15
ω3
τ
3
1
0
−1
10
−0.05
−2
5
−3
−0.1
5
10
15
τ1
20
25
−4
−4
30
−3
−2
−1
0
ω1
1
2
3
4
FIG. 10. The response function 共left兲 and the absolute value of its Fourier transform 共right兲 for the quartic potential coupled to a two-level-system bath at
ប␻0␤ = 0.1.
specified in Eq. 共2.24兲. At very low temperatures, bath excitation is negligible, thus the effects of the TLS bath should
equivalent to those of the harmonic bath 共with the same frequencies and coupling coefficients兲. Equation 共2.25兲 confirms this expectation. However, as the temperature is increased, the TLS bath begins to deviate drastically from the
harmonic one, becoming equivalent to a harmonic bath of a
very different spectral density.56
In Fig. 10 we compare the response functions of the
quartic oscillator coupled to a harmonic bath and a TLS bath
of the same frequencies and coupling coefficients. We show
results only at high temperatures, where the differences are
most pronounced. Even though coupling to the TLS bath
leads to decay in the response function, this effect is now less
dramatic compared to the effect of a harmonic bath with the
same parameters. The peak in the two-dimensional spectrum
like along the diagonals and are broader than in those observed with the harmonic bath.
V. CONCLUDING REMARKS
In this paper we have calculated nonlinear response
functions and spectra for one-dimensional model systems,
with and without coupling to dissipative baths of harmonic
oscillators and TLSs. Such four-time response functions are
relevant to third order infrared and seventh order Raman
echo experiments.
We find that the anharmonicity of the potential has a
drastic qualitative effect on the response function of these
one-dimensional models. On short time scales, even very
weak anharmonicity in a Morse oscillator leads to a 45° rotation of the response function contours, causing the peaks to
lie along the diagonals. On longer time scales, anharmonicity
results in a modulation of the amplitude of the response
function. At low temperatures this modulation is seen primarily along the ␶3 axis, but as the temperature increases, it
becomes prominent along both time axes. At high temperatures the response function becomes symmetric with respect
to the main diagonal. At high temperatures, strong anharmo-
nicity, as in the case of a potential that includes a quartic
term, can lead to extreme modulation that may give rise to
more complex patterns.
Coupling of these one-dimensional systems to dissipative environments introduces decay in the response function
and broadening in its Fourier transform. The effects of dissipation are more pronounced along the ␶3 direction. Coupling
to harmonic and TLS baths leads to similar spectra at low
temperatures where excitations are negligible. At high temperature, harmonic baths are much more effective at inducing decoherence compared to TLS baths with the same parameters.
In conclusion, the signal in nonlinear infrared and Raman experiments is very sensitive to the anharmonicity of
the system potential, coupling to the environment, and the
temperature. The patterns observed in our calculations, both
in the time domain and in Fourier space, offer useful benchmarks for identifying features of the molecular motion in
nonlinear experimental spectra.
ACKNOWLEDGMENTS
This material is based on the work supported by the
National Science Foundation under Award Nos. ITR 0427082, CHE 05-18452, and CHE 08-09699. Calculations
were performed on a Linux cluster acquired through CRIF
05-41659. We thank Ke Dong for an early version of the
computer code.
S. Mukamel, Annu. Rev. Phys. Chem. 41, 647 共1990兲.
J. T. Fourkas, Advances in Chemical Physics 共Wiley, New York, 2001兲,
Vol. 117, p. 235.
3
J. T. Fourkas, Annu. Rev. Phys. Chem. 53, 17 共2002兲.
4
O. Golonzka, M. Khalil, N. Demirdoven, and A. Tokmakoff, ACS Symp.
Ser. 820, 169 共2002兲.
5
M. D. Fayer, Annu. Rev. Phys. Chem. 60, 21 共2009兲.
6
R. M. Hochstrasser, Adv. Chem. Phys. 132, 1 共2006兲.
7
S. Mukamel, Principles of Nonlinear Optical Spectroscopy 共Oxford University Press, New York, 1995兲.
8
Y. Tanimura and S. Mukamel, J. Chem. Phys. 99, 9496 共1993兲.
9
J. Wu, J. Chem. Phys. 115, 5381 共2001兲.
10
M. Kryvohuz and J. Cao, J. Chem. Phys. 122, 024109 共2005兲.
11
R. Akiyama and R. F. Loring, J. Chem. Phys. 116, 4655 共2002兲.
12
R. Akiyama and R. F. Loring, J. Phys. Chem. 107, 8024 共2003兲.
1
2
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134506-12
13
J. Chem. Phys. 132, 134506 共2010兲
M. M. Sahrapour and N. Makri
W. G. Noid, G. S. Ezra, and G. F. Loring, J. Chem. Phys. 119, 1003
共2003兲.
14
W. G. Noid, G. S. Ezra, and G. F. Loring, J. Chem. Phys. 120, 1491
共2004兲.
15
S. M. Gruenbaum and R. F. Loring, J. Chem. Phys. 129, 124510 共2008兲.
16
Y. Tanimura and Y. Maruyama, J. Chem. Phys. 107, 1779 共1997兲.
17
T. Steffen and Y. Tanimura, J. Phys. Soc. Jpn. 69, 3115 共2000兲.
18
O. Kühn and Y. Tanimura, J. Chem. Phys. 119, 2155 共2003兲.
19
T. Kato and Y. Tanimura, J. Chem. Phys. 120, 260 共2004兲.
20
A. Ishizaki and Y. Tanimura, J. Phys. Soc. Jpn. 74, 3131 共2005兲.
21
A. Ishizaki and Y. Tanimura, J. Chem. Phys. 125, 084501 共2006兲.
22
N. Makri and D. E. Makarov, J. Chem. Phys. 102, 4600 共1995兲.
23
N. Makri and D. E. Makarov, J. Chem. Phys. 102, 4611 共1995兲.
24
N. Makri, J. Math. Phys. 36, 2430 共1995兲.
25
E. Sim and N. Makri, Comput. Phys. Commun. 99, 335 共1997兲.
26
N. Makri, J. Chem. Phys. 111, 6164 共1999兲.
27
J. Shao and N. Makri, Chem. Phys. 268, 1 共2001兲.
28
J. Shao and N. Makri, J. Chem. Phys. 116, 507 共2002兲.
29
N. Makri, Chem. Phys. Lett. 193, 435 共1992兲.
30
M. Topaler and N. Makri, Chem. Phys. Lett. 210, 448 共1993兲.
31
D. E. Makarov and N. Makri, Chem. Phys. Lett. 221, 482 共1994兲.
32
N. Makri and W. H. Miller, J. Chem. Phys. 89, 2170 共1988兲.
33
J. Echave and D. C. Clary, J. Chem. Phys. 190, 225 共1992兲.
34
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals
共McGraw-Hill, New York, 1965兲.
R. P. Feynman and J. F. L. Vernon, Ann. Phys. 24, 118 共1963兲.
J. D. Doll and D. L. Freeman, Adv. Chem. Phys. 73, 289 共1988兲.
37
N. Makri, Comput. Phys. Commun. 63, 389 共1991兲.
38
N. Makri, Annu. Rev. Phys. Chem. 50, 167 共1999兲.
39
A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and
M. Zwerger, Rev. Mod. Phys. 59, 1 共1987兲.
40
J. N. Onuchic and P. G. Wolynes, J. Phys. Chem. 92, 6495 共1988兲.
41
R. A. Marcus, Angew. Chem., Int. Ed. Engl. 32, 1111 共1993兲.
42
N. Makri, J. Phys. Chem. 103, 2823 共1999兲.
43
R. Zwanzig, J. Stat. Phys. 9, 215 共1973兲.
44
A. O. Caldeira, A. H. Castro Neto, and T. Oliveira de Carvalho, Phys.
Rev. B 48, 13974 共1993兲.
45
Y. Tanimura, AIP Conf. Proc. 503, 144 共2000兲.
46
T. Steffen and Y. Tanimura, J. Phys. Soc. Jpn. 76, 078001 共2007兲.
47
M. Kryvohuz and J. Cao, Phys. Rev. Lett. 96, 030403 共2006兲.
48
E. Jezek and N. Makri, J. Phys. Chem. 105, 2851 共2001兲.
49
N. J. Wright and N. Makri, J. Chem. Phys. 119, 1634 共2003兲.
50
J. Chen and N. Makri, Mol. Phys. 106, 443 共2008兲.
51
M. Kryvohuz and J. Cao, Phys. Rev. Lett. 95, 180405 共2005兲.
52
J. Leegwater and S. Mukamel, J. Chem. Phys. 102, 2365 共1995兲.
53
W. G. Noid, G. S. Ezra, and G. F. Loring, J. Phys. Chem. 108, 6536
共2004兲.
54
S. Mukamel, V. Khidekel, and V. Chernyak, Phys. Rev. E 53, R1 共1996兲.
55
G. Ilk and N. Makri, J. Chem. Phys. 101, 6708 共1994兲.
56
K. Forsythe and N. Makri, Phys. Rev. B 60, 972 共1999兲.
35
36
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp