Perturbative Treatment of Quantum to Classical Transition in Chiral Molecules: Dilute Phase vs. Condensed Phase

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by  Farhad Taher Ghahramani

Perturbative Treatment of Quantum to Classical Transition in Chiral Molecules:
Dilute Phase vs. Condensed Phase
Farhad Taher Ghahramani1,∗ and Arash Tirandaz1,†
1Foundations of Physics Group, School of Physics,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
We examine the dynamics of chiral states of chiral molecules with high tunneling rates in dilute
andcondensedphasesinthecontextoftime-dependentperturbationtheory. Thechiralmoleculeis
effectivelydescribedbyanasymmetricdouble-wellpotential,whoseasymmetryisameasureofchiral
interactions. The dilute and condensed phases are conjointly described by a collection of harmonic
oscillators but respectively with temperature-dependent sub-ohmic and temperature-independent
ohmic spectral densities. We examine our method quantitatively by applying the dynamics to
isotopic ammonia molecule, NHDT, in an inert background gas (as the dilute phase) and in water
(asthecondensedphase). Asdifferentspectraldensitiesimplies,theextensionofthedynamicsfrom
thedilutephasetothecondensedphaseisnottrivial. Whilethedynamicsinthedilutephaseleads
to racemization, the chiral interactions in the condensed phase induce the quantum Zeno effect.
7
Moreover,contrarytothecondensedphase,theshort-timedynamicsinthedilutephaseissensitive
1
to the initial state of the chiral molecule and to the strength of the coupling between the molecule
0
and theenvironment.
2
n PACSnumbers: 33.55.+b,33.80.-b,87.10.-e,87.15.B-
a
J
I. INTRODUCTION
2

]
h One of the fundamental problems in molecular science is the quantum-mechanical origin of chirality. The chiral
p configurationsofachiralmoleculehaveamany-particle,complexdynamics. Thisdynamicscanbeeffectivelymodeled
- by the motionof a particle in a symmetric double-wellpotential [1]. The states associatedto chiralconfigurationsare
t
n assumed to be localized in two minima of the potential. The superposition of chiral states is realized by tunneling
a through the barrier. If the barrier is high enough to prevent tunneling, once prepared, the chiral state is preserved.
u
Upon a quantitative analysis, however, this explanation seems rather insufficient for low-barrier molecules [2]. As a
q
[ modern resolution, the chiral interactions, i.e. interactions that prefer a particular chiral state, became important.

They can be effectively taken into account by introducing an asymmetry parameter into the symmetric double-well
1
potential. If they are strong enough to overcome tunneling process, the molecule is confined in the preferred chiral
v
5 state. The most discussed interactions in this context are parity-violating weak interactions [3, 4], intermolecular
3 interactions [5] and interaction with the circularly-polarized light [6]. The problem is that the induced chirality is
5 demolished by the dissipative effects of the surrounding environment. The open chiral molecule has been studied by
0
the mean-field approach, and more completely by the decoherence theory. In the mean-field theory, the environment
0
is envisaged as an effective potential added to the Schr¨odinger dynamics of the system [7]. The resulting non-linear
.
1 dynamics can be served to stabilize a particular chiral state [8–10]. Recently, this approach is extended to include
0
chiral interactions in the Langevin formalism of open quantum systems [11–15].
7
Thedecoherencetheorydescribethesuperselectionofapreferredsetofsystem’sstatesbytheenvironment-induced,
1
: dynamical destruction of quantum coherence between them [7, 16]. The environmentacting upon the molecule might
v
be dilute (e.g., gaseous environment) or condensed (e.g., biological environment). The dynamics of a system in a gas
i
X phase was studied by the scattering model or more conveniently collisional decoherence [17–20]. The application of
r collisionaldecoherencetothestudyofthetunnelingofachiralmoleculeinagasphasewasfirstdoneinthepioneering
a
works of Harris and Stodolsky [21, 22] and Silbey and Harris [23]. More refined works showedthat the environmental
collisions induce an indirect position-measurement on the molecule and thus stabilize the chiral states [24–28]. In a
condensed phase, however, since the medium is always present, the idea of a collision is inapplicable. The simplest
representation of a condensed phase is a bath of harmonic oscillators. At sufficiently low temperatures, the states of
the molecule areeffectively confinedinthe two-dimensionalHilbert spacespannedby the two chiralstates. When the
two-levelsystemcoupleslinearlytotheenvironmentaloscillators,theresultistherenownedSpin-Bosonmodelstudied
extensivelyintheliterature,especiallybyLeggettandco-workers[29]. Recently,thismodelisemployedtoexaminethe
role of quantum coherence in biological systems [30, 31]. An elementary application of this model to chiral molecules
isfoundintheworkofSilbeyandHarris[23]. Recently,Tirandazandco-workerssolvedthegeneralSpin-Bosonmodel
to examine the interplay between tunneling process and chiral interactions in a biological environment [32]. They
showedthat despite the powerful racemizationeffects of the biologicalenvironmentthe chiralinteractions induced by
the same environment can prohibit the tunneling process and thus preserve the initial chiral state.
Thedescriptionofthedecoherencedynamicsisusuallyformulatedintermsoftheso-calledmasterequations,which
yield the approximate evolution of the reduced density matrix of the open quantum system. The master equation is
essentially a mathematicalobjectwhich maps the initial state to the finalstate without providinga clear-cutphysical
understanding of the inner dynamics. Moreover, it is recently discussed that conceptually the state of the closed
2
system andthe reducedstate assignedto that systemwhen it interactswith the environmentare different[33, 34]. In
order to present a simple and straightforwarddemonstration of the stabilization of molecular chirality, in this paper,
we apply the time-dependent perturbation theory to examine the dynamics of an open chiral molecule. The main
emphasis here would be on the comparisonbetween dynamics influenced by dilute and condensed environments. It is
demonstrated that the interaction with any environment can be rigorouslymapped onto a system linearly coupled to
an oscillator environment, provided the interaction is sufficiently weak and second-order perturbation theory can be
applied [35, 36]. We describe the dynamics of chirality by an asymmetric double-well potential, and the environment
is representedbya collectionofharmonicoscillators,withdifferentspectraldensities for dilute andcondensedphases.
We exemplify the dilute and condensed phases respectively by an inert background gas and water. Note that for we
focus on chiralmolecules with high tunneling rates, the strength of chiralinteractions can be consideredless than the
tunneling strength, especially in the dilute phase.
Thepaperisorganizedasfollows. Inthenextsection,wedescribethechiralmoleculeanditsharmonicenvironment.
We examine the dynamics of chiral states in the third section using time-dependent perturbation theory. In the forth
section,wefirstestimatetheparametersrelevanttoouranalysis,specifythespectraldensitiesofdiluteandcondensed
environments and accordingly present our results. Finally, we summarize our findings in the last section.
II. MODEL
The total Hamiltonian of the entire system composed of the chiral molecule and the environmental particles is
conveniently defined as
H =H +H +H (1)
M E ME
where H , H and H are the molecular, environmental and interaction Hamiltonians, respectively.
M E ME
A chiral molecule is found at least as two identical enantiomers through the inversion at molecule’s center of mass
by a long-amplitude vibration known as the contortional vibration [37, 38]. This particular vibration can effectively
be described by the motion of a particle in an asymmetric double-well potential. The minima of the potential are
associatedtotwoenantiomersofthemoleculeandtheasymmetrycanbe consideredasanoverallmeasureofallchiral
interactions. For an isolated chiral molecule, the asymmetry is merely resulted from the internal chiral interactions
i.e., the parity-violatingweak interactions[3, 39]. For a chiralmolecule in interactionwith the environment,however,
the external chiral interactions e.g., the dispersion intermolecular interactions [5] and interaction with the circularly-
polarized light [28] are the main source of asymmetry. An asymmetric double-well potential can be represented by a
quartic potential including a linear term
U(R)=U R 2 1 2 1 η R , U = MΩ2R◦2 (2)
◦ R − − − R ◦ 8
◦ ◦
nh(cid:0) (cid:1) i (cid:0) (cid:1)o
where M is the effective mass of the chiral molecule, Ω is the harmonic frequency at the bottom of each well, η is
calledtheasymmetryparameterandR isthedistancebetweentwominimafromtheoriginforη =0. Toquantifythe
◦
extent to which the molecule exhibits quantum coherence, we incorporate the dimensionless form of the model. The
potential has the characteristic length R and the characteristic energy U which we adopt as the units of length and
◦ ◦
energy,respectively. Thecorrespondingcharacteristictimecanbedefinedasτ =R /(U /M)1/2 whichweconsideras
◦ ◦ ◦
the unit of the time. Likewise, the unit of the momentum is taken as P =(MU )1/2. We then define the dynamical
◦ ◦
variables,x and p, as R/R and P/P , respectively. The correspondingcommutation relationis defined as [x,p]=ıh,
◦ ◦
where Planck constant is redefined as h=~/R P =~/U τ , which we call the reduced Planck constant.
◦ ◦ ◦ ◦
In the limit E Ωτ h 1 (with E = k T/U as thermal energy, k as Boltzmann constant, and T as
th ≪ ◦ ≪ th B ◦ B
temperature), energy states are confined in two-dimensional Hilbert space spanned by two chiral states L and R .
| i | i
In fact, such a two-level approximation holds for most chiral molecules [37, 38]. Accordingly, the effective molecular
Hamiltonian in the chiral basis can be written as [29]
H = ∆σ δσ (3)
M x z
− −
where we defined
hΩτ
∆= ◦
4
h
δ =η (4)
2Ωτ
r ◦
inwhich∆isthetunnelingstrengthandδ,knownasthelocalizationstrength,isameasureofthetheenergydifference
between two enantiomers. If chiral interactions are strong enough to overcome tunneling process, ∆ δ, the chiral
≪
states become stable, and thus the chirality problem is resolved. Therefor, here we focus on chiral molecules with
3
low barrier, where the tunneling strength is larger than the localization strength, ∆ δ. The states of energy are
≫
described by the superposition of localized states as
1 =cosθ R +sinθ L
| i | i | i
2 =sinθ R cosθ L (5)
| i | i− | i
wherewedefinedθ = 1arctan ∆/δ . Thecorrespondingenergiesare 1(∆2+δ2)1/2. Foranisolatedchiralmolecule,
2 { } ∓2
the probability of the tunnelling from the left state to the right one is given by
∆2
P = sin2 (∆2+δ2)1/2t/2 (6)
L→R ∆2+δ2 { }
A frequently employed model for an environment is a set of harmonic oscillators. The α-th harmonic oscillator in
the environment is characterized by its natural frequency, ω , and position and momentum operators, x and, p ,
α α α
respectively, according to the Hamiltonian
1 1 1
H = p2 + ω2x2 hω (7)
E 2 α 2 α α− 2 α
Xα (cid:16) (cid:17)
The last term, which merely displaces the originof energy, is introduced for later convenience. We define vac as the
| i
vacuum eigenstate and α as an excited eigenstate of the environmental Hamiltonian with energy ε .
α
| i
The interaction Hamiltonian is generally defined by a linearly-coupled harmonic-environmentmodel as
1
H = ω2f (σ )x + ω2f2(σ ) (8)
ME − α α z α 2 α α z
Xα (cid:16) (cid:17)
where the function f (σ ) is the displacement in each environmental oscillator α induced by the molecule. For
α z
simplicity, we assume that the interaction model is separable (f (σ ) = γ f(σ ), where γ is the coupling strength)
α z α z α
and also bilinear (f(σ )=σ ).
z z
The shift in the molecular energy due to the perturbation H is obtained up to the second order as
ME
m,αH n,vac 2
ME
δE n,vacH n,vac + |h | | i|
n ME
≃h | | i E (E +ε )
n m α
m6=n −
X
α6=vac
1 γ2ω3
= σ 2Ω α α (9)
rn rn
2 | | ω (ω +Ω )
α α rn
r α
X X
where σ = mσ n , Ω = E E /h. The state with the energy shifted to E +δE by perturbation is not
mn z mn m n n n
h | | i −
actually stationary and decays with a finite lifetime Γ−1 given by Fermi’s golden rule as
n
2π
Γ n,vacH n,vac 2δ(E (E +ε ))
n ME n m α
≃ h |h | | i| −
m6=n
X
α6=vac
π γ2ω3δ(Ω ω )
= σ 2Ω α α rn− α (10)
rn rn
h | | ω
α
r α
X X
III. DYNAMICS
We assume that the initial state of the entire system can be written as
Ψ(0) = ψ vac (11)
| i | i| i
where ψ is an arbitrary state of the chiral molecule. The state of the total system at time t is obtained by
| i
Ψ(t) =e−ıH◦t/hU (t)Ψ(0) (12)
I
| i | i
where we defined H = H + H and U is the time evolution operator in the interaction picture: U (t) =
◦ M E I I
eıH◦t/he−ıHt/h. We expand U (t) up to the second order with respect to the interaction Hamiltonian to find
I
ı t 1 t t2
U (t) 1 dt H (t ) dt dt H (t )H (t ) (13)
I ≃ − h 1 ME 1 − h2 2 1 ME 2 ME 1
Z0 Z0 Z0
4
First, we calculate
U (t)vac U (t)vac +U (t)α +U (t)αβ (14)
I vac α αβ
| i≃ | i | i | i
where
ıt 1 t t2
U (t)=1 ω2γ2 ω3γ2 dt dt e−ıωα(t2−t1)σ (t )σ (t )
vac − 2h α α− 2h α α 2 1 z 2 z 1
α α Z0 Z0
X X
ı t
U (t)= ω3/2γ dt eıωαt1σ (t )
α √2h α α 1 z 1
Z0
1 t t2
U (t)= dt dt ω3/2ω3/2γ γ eı(ωαt1+ωβt2)σ (t )σ (t ) (15)
αβ −2h 2 1 α β α β z 2 z 1
Z0 Z0
to find
2
ψ(t) = e−ıEnt/h n vac nU (t)ψ + e−ıωαt α nU (t)ψ + e−ı(ωα+ωβ)t αβ nU (t)ψ (16)
vac α αβ
| i | i | ih | | i | ih | | i | ih | | i
nX=1 n Xα Xαβ o
wherewedefinedσz(t)=eıHMt/hσze−ıHMt/h. Thecontributionofthethirdtermof(16)intheprobabilityoftunneling,
being of order γ4, and hence beyond the second order, can be securely dropped. The problem is thus reduced to the
evaluation of matrix elements of U (t) and U (t). The diagonal matrix elements of U (t) are evaluated as
vac α vac
ıt ı ∞ t 1 e−ı(ω+Ωmn)t′
nU (t)n =1 δE(1)+ σ 2 dω J(ω) dt′ − (17)
h | vac | i − ~ n πh | mn| ω+Ω
m Z0 Z0 mn
X
where J(ω) is the spectral density of the environment, corresponding to a continuous spectrum of environmental
frequencies, ω, defined as
π
J(ω)= γ2ω3δ(ω ω ) (18)
2 α α − α
α
X
In the third term of (17), the result of the integration remains unchanged if the integral is regarded as the principal-
value integral which allows us to perform the t′-integration term by term as
ıt 1 ∞ J(ω) 1 ∞ 1 e−(ω+Ωmn)t
nU (t)n =1 δE(1) σ 2℘ dω σ 2℘ dωJ(ω) − (19)
h | vac | i − h( n − π m | mn| Z0 ω+Ωmn)− πh m | mn| Z0 (ω+Ωmn)2
X X
where the symbol ℘ denotes the principal-value integral. The quantity embraced by the braces in the second term on
the right hand side coincides with δE in (9). So, the following expression is valid up to the second order
n
hn|Uvac(t)|ni≃e−πıthδEn(1− π1h m |σmn|2 ℘Z0∞dωJ(ω)1−(ωe+−(Ωω+mΩnm)2n)t)
X
1 ∞ sin (ω+Ω )t/2 2 sin (ω+Ω )t
=e−πıthδEn(1− πhXm |σmn|2"℘Z0 dωJ(ω)h2(cid:16) {ω+Ωmrnn }(cid:17) −ı ({ω+Ωmmn)n2 }i#) (20)
Weassumethatenvironmentalcut-offfrequencyΛismuchhigherthanthecharacteristicfrequencyofthemoleculeΩ,
soattimes muchhigherthanΩ−1,weapproximatethe firsttermofthe integralonthe secondlinebyadelta function
δ(ω+Ω ). The result of the corresponding integral would be J(ω+Ω ), which is zero if Ω 0. Moreover, in
mn mn mn
≥
the limit ∆ δ, we can safely approximate σ 0. The diagonal elements of U (t) are then reduced to
nn vac
≫ ∼
ısin2(2θ) ∞ sin (ω Ω )t )
1U (t)1 e−ıtδE1/h 1 ℘ dωJ(ω) { − 12 } (21)
h | vac | i≃ ( − πh Z0 (ω−Ω12)2 )
and
1 ısin2(2θ) ∞ sin (ω+Ω )t )
2U (t)2 e−ıtδE2/h 1 Γ t ℘ dωJ(ω) { 12 } (22)
h | vac | i≃ ( − 2 2 − πh Z0 (ω+Ω12)2 )
where Ω = (∆2+δ2)1/2. The off-diagonalelements of U can be approximated as zero in the limit ∆ δ.
12 vac
− ≫
Matrix elements of U (t) are obtained by
α
ı sin (ω +Ω )t/2
mU (t)n = ω3/2γ σ { α mn }eı(ωα+Ωmn)t/2 (23)
h | α | i √2h α α mn (ωα+Ωmn)/2
5
To obtain the explicit forms of matrix elements of (23), one can follow a procedure pretty much the same as above.
Especially, one can show that the diagonal elements of U (t) can be estimated as zero in the limit ∆ δ.
α
≫
We suppose that the initial state of the molecule is the left-handed state L . The evolved state function of the
| i
whole system in the chiral basis of the molecule can be written as
Ψ(t) = sinθ χ (t) cosθ χ (t) L + cosθ χ (t) +sinθ χ (t) R (24)
1 2 1 2
| i | i− | i | i | i | i | i
n o n o
with
χ (t) =e−ıEit/h vac iU (t)L + e−ıωαt α iU (t)L (25)
i vac α
| i | ih | | i | ih | | i
n Xα o
for i=1,2. We are interested in the probability of finding the molecule in the right-handed state, i.e.,
P = RΨ(t) 2 =cos2θ χ (t)χ (t) +sin2θ χ (t)χ (t) +sin(2θ) ( χ (t)χ (t) ) (26)
R 1 1 2 2 1 2
|h | i| h | i h | i ℜ h | i
IV. RESULTS
To examine the effective dynamics of the open chiral molecule, we first estimate the parameters relevant to our
analysis. Twoextremecasescanbedistinguished: chiralmoleculeswithverylongtunnelingtimesandchiralmolecules
with rapid tunnellings. For former, we have δ ∆, giving θ 0, chiral states become stationary states and Hund’s
≫ →
Paradox is resolved [40, 41]. So, here we just focus on latter. The free parameters of the molecule can be chosen to
be the effective mass M,the characteristicenergyU and the characteristiclength R . The effective mass of a typical
◦ ◦
small chiral molecule (e.g., isomeric ammonia, NHDT) is estimated as M 10−27Kg. The two-level approximation
∼
requires that E Ωτ h 1, so considering the thermal energy at the room temperature as k T 10−21J, we
th ≪ ◦ ≪ B ∼
estimate the typical value of characteristicenergyas U 10−19J. We estimate the characteristiclength for a typical
◦
∼
molecule close to molecular lengths as R 10−10m. The restofthe molecular parametersareestimated accordingly.
◦
∼
We summarize relevant molecular parameters in TABLE I.
Parameter Value Parameter Value Parameter value
M 10−27Kg P◦ 10−23Kg.m.s−1 Ω 1013Hz
U◦ 10−19J τ◦ 10−14s ∆ 10−3
R◦ 10−10m h 0.1 δ η
TABLE I: Molecular parameters relevant for themodel.
Note that the value of tunneling frequency ∆ coincides with the tunneling frequency of NHDT. Of particular interest
is the value of reduced Planck constant obtained as h 0.1. The parameter h quantifies the extent to which the
∼
isolatedmolecule exhibit quantum coherence. The situation in whichh 1 is called the quasi-classical situation[42].
≪
Foranisolatedchiralmolecule,theprobabilityoftheright-handedstate,accordingto(6),aspreviouslyreportedby
HarrisandStodolsky[21,22]andrecentlybyBarguen˜oandco-workers[11],isimplicitlyindependentofreducedPlanck
constanth. Atthetunneling-dominantlimit,correspondingtothesymmetricdouble-wellpotential,thedynamics(blue
plot in FIG. 1,a) shows symmetric oscillations between two chiral states.
(a) (b)
FIG. 1: The isolated chiral molecule: a) dynamics of the right-handed state for η=10−4 (blue), η=10−3 (brown), η =10−2
(green) b) probability of the right-handedstate versus theasymmetry parameter η at t=1000.
6
Introducing the asymmetry into the unitary dynamics reduces the symmetry of the oscillations, eventually confining
themtotheinitialchiralstate(brownandgreenplotsinFIG.1,a). Thisoscillatorydynamicsisthequantumsignature
of the system. At a definite time, the probability of the right-handed state is decreased quickly with the asymmetry
parameter, η, (FIG. 1,b). This is because the chiral interactions overcome tunneling process, and thus confine the
molecule in the initial chiral state.
A molecule, especially in a biologicalenvironment,is not really isolated. The environmentaleffects on the molecule
may be classified into decoherence and dissipation. The former is the suppression of the relative phases (coherences)
between the states of the molecule. The decoherence in the energy basis of the molecule is usually called dephasing.
In our approach, the decoherence effect is characterized by the third term of (26). The latter is the energy exchange
with the environment leading to thermalization which is usually accompanied by decoherence. In our approach, first
and second terms of (26) are responsible for dissipation effect. In order to examine the dynamics of the open chiral
molecule, first we should specify the type of the environment.
Dilute Environment
The interaction of the molecule with a dilute environment can be approximately accounted by two-body collisions.
A gas phase is a generic example of such an environment. A gas phase is not actually a collection of harmonic
oscillators. In fact, the fluctuating force produced by the gas particles onto molecule is not Gaussian (as it would
be for oscillators), but rather looks like a series of random delta peaks. Nevertheless, if every collision is sufficiently
weak, then fluctuating force could still approximately consideredas Gaussian(when averagedover longer time scales,
including many collisions). For the gas phase we obtain a simple expression for the right-handed state dynamics.
For the coupling between the molecule and gas particles is weak, we have Γ Ω. At the temporal domain
2
Ω−1 t Γ−1, we obtain ≪
≪ ≪ 2
cos2(θ)sin2(2θ) ∞ sin2 (ω+Ω )t )
P (t)= χ (t)χ (t) sin2(θ)+ ℘ dωJ(ω) { 12 }
1 h 1 | 1 i≃ πh (ω+Ω )2
Z0 12
=sin2(θ)+cos2(θ)Γ t sin2(θ)+cos2(θ)(1 e−Γ2t) (27)
2
≃ −
and
sin2(θ)sin2(2θ) ∞ sin2 (ω Ω )t )
P (t)= χ (t)χ (t) cos2(θ)e−Γ2t+ ℘ dωJ(ω) { − 12 }
2 h 2 | 2 i≃ πh (ω Ω )2
Z0 − 12
cos2(θ)e−Γ2t (28)
≃
One can easily show that cos2(θ)P (t)+sin2(θ)P (t) 1/2 and P (t)P (t) 1. So, the effect of the environment
1 2 1 2
∼ ∼
manifests itself through the dephasing factor χ (t)χ (t) rather than the dissipation factors P (t) (i=1,2). This is
1 2 i
h | i
in accordancewith the corresponding master equation in the dilute-phase limit of collisional decoherence [27, 43, 44].
The dephasing factor is obtained as
sin(2θ)
χ (t)χ (t) e−Γ2t/2exp ıΩ t+ı(δE δE )t
1 2 21 2 1
h | i≃ 2 ( " −
ısin2(2θ) ∞ sin (ω Ω )t sin (ω+Ω )t
12 12
+ dω J(ω) { − } { }
πh Z0 (cid:16) ω−Ω12 − ω+Ω12 (cid:17)#
4ısin2(2θ) ∞ sin (ω Ω )t sin (ω+Ω )t
12 12
+ dωJ(ω) { − } { } (29)
πh ω2 Ω2
Z0 − 12 )
Finally, the right-handed probability in the temporal domain Ω−1 t Γ−1 is reduces to
≪ ≪ 2
sin2(2θ)
P cos2(θ)P +sin2(θ)P e−Γ2t/2cos (Ω +δE δE )t+ζ (30)
R 1 2 21 2 1
∼ − 2 −
n o
where
sin2(2θ) J(Ω )
ζ = J′(Ω ) 21 (31)
21
h − Ω
21
(cid:16) (cid:17)
where we defined J′(ω)=dJ(ω)/dω. The dephasing effects of the gas phase are manifested in e−Γ2t/2 and ξ.
In order to examine the dynamics explicitly, we should specify the spectral density of the gas phase. As Harris and
Stodolsky demonstrated, the relevant dynamics of chiralstates samples the velocity distribution of the gas molecules,
7
which is strongly temperature-dependent [21, 22]. Starting from a microscopic model of the collisions with the gas
particles, we derive the spectral density
∞
J(ω)= dt X(t)X(0) eıωt (32)
E
h i
Z−∞
whereX(t)istheinteraction-picturepositionoperatoroftheenvironmentalparticles. Notethatforsimplicitywedrop
all normalization factors expressed by 2π. The interaction process envisaged here is a sequence of collisions between
gas particles and a heavier chiral molecule. We also assume that the collision doesn’t lead to any internal transition
of the molecule. The position operator of the environmental particles can be expanded as [45]
X = d3r a(r)ρ¯ (r) (33)
E
Z
where a(r) is the interaction function and ρ¯ (r) is the difference between the gas density operator ρ (r) and its
E E
time-averagedvalue, assumed to be the uniform gas density ρ. If we substitute (33) in (32), we have
J(ω)= d3q a˜(q)2S(q,ω) (34)
| |
Z
where a˜(q) is the Fourier transform of the interaction function
a˜(q)= d3r a(r)e−ıq.r (35)
Z
and we defined
S(q,ω)= dt d3r eı(ωt−k.r) ρ¯ (0,0)ρ¯ (r,t) (36)
E E
h i
Z Z
which is essentially the spectral density appeared in the first order theory of scattering [46]. The density operator of
a gas of free particles is conveniently defined as
1
ρ (r)= eıq.rb† b (37)
E V k+q k
kq
X
where V is the normalization volume and b is the annihilation operator. If we insert (37) into (36), we obtain
k
E E
S(q,ω)=ρ d3p n¯ (1 n¯ )δ(ω p− hk) (38)
p p−hk
± − h
Z
Here,ρisthedensityofthegasparticlesperunitvolume,n¯ = b†b istheBoseorFermidistributionandE =p2/2
hk h k ki p
is the single-particle energy. We are interested in the classical, non-degenerate regime, i.e, n¯ 1, which leads to
p
n¯p(1 n¯p−hk) e−p2/2Eth. After some mathematical manipulation, the spectral density become≪s
± ≈
ρt ∞ a˜(q)2 (ω2t2c + t2Qq2)
J(ω)= ceωtQ | | e− q2 t2c (39)
R q
Z0
where R is the range of intermolecular interaction,t = (R2/2E )1/2 and t = h/E are the classical and quantum
c th Q th
correlationtimes, respectively. Atroomtemperature,t 10−1, andforaninteractionrangeR>1andgasparticles
Q
≈
heavier than Hydrogen molecules, we estimate t > 1. Thus, we ignore the term t2/t2 (< 10−2). For a gaussian
c Q c
interaction function a˜(q)2 =e−q2, and assuming t t , the explicit form of the spectral density would be
c Q
| | ≫
J(ω)=J ω21e−Λω (40)
◦
with coupling strength J ρE−3/4 and cut-off frequency Λ = 2(4t t )−1. The spectral density (40) shows a
◦ ∝ th c − Q
sub-ohmic frequencydependence witha temperature-dependentcoupling strengthandcut-offfrequency. Atthe room
temperature, assuming the dilute gas limit, we estimate J <1, and Λ>0.5.
◦
Condensed Environment
Inacondensedenvironment,sincethemediumisalwayspresent,theideaofcollisionisinapplicable. Theproperties
of such an environment can be encapsulated in an ohmic spectral density with an exponential cutoff Λ as
J(ω)=J ωe−Λω (41)
◦
8
in which J is a measure of the system-environmentcoupling strength. A condensedenvironmentis more appropriate
◦
to represent a biological environment. The interaction of the molecule with a biological environment is, by nature,
complex. We assume that the chiral molecule is surrounded by a uniform polar solvent. For simplicity, we describe
the solvation process by the well-known Onsager model [47]. The chiral molecule is treated as a point dipole which
is surrounded by a spherical cage of polar solvent molecules with Onsager radius a, which is typically the size of the
molecule. For a Debye solvent, the parameters of the spectral density can be written as [48]
(∆µ)2 6(ǫ ǫ )
s ∞
J = − (42)
◦ 4πǫ a3(2ǫ +1)(2ǫ +1)Λ
◦ s ∞
and
1 2ǫ +1
s
Λ= (43)
τ 2ǫ +1
D ∞
where∆µisthedifference betweenthedipole momentofthe moleculeinthe groundandexcitedstates,ǫ andǫ are
s ∞
the static and high-frequency dielectric constants of the solvent, and τ is the Debye relaxation time of the solvent.
D
For water as the most prevalent solvent we have ǫ = 78.3, ǫ = 4.21 and τ = 8.2ps [49]. Using these parameters,
s ∞ D
we obtain (dimension-less) cut-off frequency as Λ 0.01. Note that if Λ Ω , the environmental particles cannot
12
≈ ≪| |
resolve the molecular states and thus the quantum oscillations resulted from tunneling process preserves. Also, if we
measure the cavity size and change in dipole moment in angstroms and Debye respectively, then we obtain coupling
strength as J = 22(∆µ)2. For a typical small molecule with radius 1˚A, a dipole moment change of just 0.2D is
◦ a3 ≈ ∼
sufficient to make J >1, and this condition seems likely to be met for most small molecules [50]. As an example, for
◦
NHDT with ∆µ 0.6D [51], we have J 10.
◦
≈ ≈
Now, we compare the dynamics in a dilute phase with that of a condensed phase at short- and long-time limits.
Long-Time Dynamics The equilibrium state is obviously reachedin the dilute and condensed phases at different
rates. The condensed-phase equilibrium, as expected, occurs faster than the dilute-phase one (FIG. 2). Of course,
the value of the rates depends on the details of the corresponding spectral density. For NHDT molecule in hydrogen
gas and water, the equilibrium is reached after 10000 (10−11s) and 1000 (10−10s), respectively. In the dilute
∼ ∼
phase, the chiral interactions are relatively weak and the chiralmolecule in question has a high tunneling rate, so the
dynamicsis confinedto the tunneling-dominantlimit whereδ ∆. Inthe condensedphase,however,the short-range
≪
intermolecularchiralinteractionswouldbe significant. Atthe tunneling-dominantlimit, inbothdilute andcondensed
phases, the equilibrium state is a racemic mixture (FIG. 2,a and blue curve of FIG. 2,b). Introducing the asymmetry
into the condensed-phase dynamics leads to the localization. In the condensed phase, the chiral interactions confine
the molecule in the initial chiral state (green curve of FIG. 2,b). These results are in agreement with those of master
equationapproach( for dilute phase [27], for condensedphase [32]), and Langevinapproach[11] for condensedphase.
In both phases, the equilibrium state is independent of the chirality of the chiral interactions (FIG. 2). Also, the
dilute- and condensed-phase dynamics in our work is compatible respectively with the weak- and strong-damping
regimes of Spin-Boson approach[52, 53]. Note that the equilibrium state, as Grifoni and co-workerspredicted for the
drivenSpin-Bosonmodelusingthepathintegralmethod[54],isindependentofthe initialstateofthe chiralmolecule.
Obviously, the rate at which the equilibrium state is reached in both phases depends on the coupling strength J . In
◦
the dilute phase,as expected,the equilibriumrateis increasedwiththe couplingstrength. However,in the condensed
phase,theequilibriumrateisapproximatelyindependentofthecouplingstrength(FIG.3). Thisisbecauseofthefact
that the the number of collisions in the condensedphase are already reachedits asymptotic value and thus increasing
the couplingstrengthdoesn’tchangethe equilibriumrate. The equilibrium ratealsodepends onthe cut-offfrequency
of the environment Λ. In both phases, the equilibrium rate is decreased with the cut-off frequency (FIG. 4).
PR PR
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
t t
0 5000 10000 15000 20000 0 1000 2000 3000 4000 5000
(a) (b)
FIG. 2: The dynamics of the right-handed state of the open chiral molecule a) in the dilute phase at |δ| = 10−5 for δ > 0
(blue) and δ<0 (orange) b) in thecondensed phase for δ=±10−5 (blue), ±10−4 (orange) and ±10−3 (green).
9
PR PR
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
t t
0 5000 10000 15000 20000 0 1000 2000 3000 4000 5000
(a) (b)
FIG. 3: Thedynamicsof theright-handedstate oftheopenchiral molecule at |δ|=10−5 a) in thedilutephasefor J◦ =10−4
(blue), 10−3 (orange) and 10−2 (green) b) in thecondensed phasefor J◦ =10 (blue),20 (orange) and 30 (green).
PR PR
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
t t
0 5000 10000 15000 20000 0 500 1000 1500 2000
(a) (b)
FIG. 4: The dynamics of the right-handed state of the open chiral molecule at |δ|=10−5 a) in the dilute phase for Λ=10−1
(blue), 10−2 (orange) and 10−3 (green) b) in thecondensed phasefor Λ=10−2 (blue),10−1 (orange) and 1 (green).
The short-time dynamics The short-time dynamics in the dilute phase is oscillatory and strongly dependent on
the chiralityofthe chiralinteractions(FIG. 2,a). Inthecondensedphase,however,the quantumcharacteristicsofthe
molecule are completely suppressed by to the strong interactions with the environment (FIG. 2,b).
V. CONCLUSION
The dynamics of chiral states of chiral molecules is conveniently examined in the dilute and condensed phases,
respectively, by scattering approach (collisional decoherence) and Born-Markov master equation. Two approaches
being mathematically different, the extension of the molecular dynamics from the dilute phase to the condensed
phase is not straightforward. Here, we examined the dynamics of chiral states, corresponding to the left- and right-
handed states ofanasymmetric double-wellpotential, inboth phases by a unified approachusing the time-dependent
perturbation theory. Two phases are described as a harmonic environment. The spectral density of the dilute phase,
exemplified by an inert background gas, is temperature-dependent and sub-ohmic (40), while the spectral density of
the condensed phase, manifested as water, is temperature-independent and ohmic (41). As our analysis implies, the
dynamics in the dilute-phase cannot be extended to the dynamics of the condensed phase by increasing the coupling
strength. This suggestthatin the condensedphase,the unexpected featuresmay emergeinthe dynamics. Especially,
we showed that the chiral interactions in the condensed phase may induce localization in the sense of the quantum
Zenoeffect(FIG.2,b),whilethedilute-phasedynamicsalwaysleadstoracemization(FIG.2,a). Also,intheshort-time
limit, the dilute-phase dynamics strongly depends on the initial state of the molecule (FIG.2,a) and on the coupling
strength (FIG.3,a), whereas the condensed-phase dynamics is not. Our results are compatible with the results of the
relevant works in the collisional decoherence, the master equation and Langevin approaches.
10
ACKNOWLEDGEMENT
F.T.G acknowledges the financial support of Iranian National Science Foundation (INSF) for this work.
∗ Corresponding Author:[email protected]
† [email protected]
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