Dissipative geometric phase and decoherence in parity-violating … · 2016. 6. 8. · Within a generalized Langevin framework for open quantum systems, the cyclic evolution of a

Dissipative geometric phase and decoherence in parity-violating chiralmoleculesA. Dorta-Urra, H. C. Peñate-Rodríguez, P. Bargueño, G. Rojas-Lorenzo, and S. Miret-Artés Citation: J. Chem. Phys. 136, 174505 (2012); doi: 10.1063/1.4707735 View online: http://dx.doi.org/10.1063/1.4707735 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i17 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 136, 174505 (2012)

Dissipative geometric phase and decoherence in parity-violatingchiral molecules

A. Dorta-Urra,1 H. C. Peñate-Rodríguez,2,3 P. Bargueño,4,a) G. Rojas-Lorenzo,2,3

and S. Miret-Artés31Unidad asociada UAM-CSIC, Instituto de Física Fundamental (CSIC), Serrano 123,E-28006 Madrid, Spain2Instituto Superior de Tecnología y Ciencias Aplicadas, Ave. Salvador Allende y Luaces,Quinta de Los Molinos, Plaza, La Habana 10600, Cuba3Instituto de Física Fundamental (CSIC), Serrano 123, E-28006 Madrid, Spain4Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain

(Received 18 January 2012; accepted 12 April 2012; published online 3 May 2012)

Within a generalized Langevin framework for open quantum systems, the cyclic evolution of a two-level system is analyzed in terms of the geometric phase extended to dissipative systems for Ohmicfriction. This proposal is applied to the dynamics of chiral molecules where the tunneling and parityviolating effects are competing. The effect of different system-bath coupling functions in the dissi-pated energy is shown to be crucial to understand the behavior of the geometric phase as well asthe decoherence displayed by the corresponding interference patterns. © 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4707735]

I. INTRODUCTION

Nowadays measuring the energy difference betweenenantiomers of chiral molecules is considered as one of themost challenging experiments.1 After more than 40 yearsof research, the parity-violating energy difference (PVED)between the two enantiomers of chiral molecules (a conse-quence of the theory of electroweak interactions), still re-mains elusive for an experimental detection. This PVED isestimated to be between 10−13 and 10−21 eV (for recent re-views see Refs. 2–7) but no conclusive energy difference hasbeen reported up to date.8–10 Due to the weakness of theeffect, extremely precise theoretical calculations are neededto propose appropriate molecular candidates to guide theexperiment.11–14 Since the first experimental suggestion,15

several techniques have been proposed for the observationof parity violation in molecular systems such as, for ex-ample, laser spectroscopy on vibrational transitions,8, 10, 14, 16

electronic,17, 18 Mossbauer9, 19 and NMR spectroscopies,20, 21

and optical activity measurements.22–35 In the last type ofmeasurements, time dependent optical activity is proposed inchiral molecules creating superpositions of eigenstates of thetwo asymmetric wells, where the tunneling and PVED con-tributions are of the same order of magnitude.22, 27 Thus, suchexperiments would provide indirect information on the PVEDby comparison with the tunneling splittings (for an exhaustivelist of tunneling splittings and PVEDs, see Table 6 of Ref. 7and references therein).

Essentially, the quantum beating of the optical activityis a consequence of the oscillation between left- and right-handed states, this being a typical example of a cyclic evo-lution. A well known consequence of quantum mechanics isthat the initial and final state vectors of a cyclic evolution

a)[email protected].

are related by a phase factor with observable consequences,two paradigmatic examples being the Berry36 and Aharonov-Anandan phases.37 Although geometrical phases have beenmeasured in many different fields of physics,38 in molecu-lar physics they have been mainly studied in the context ofconical intersections,39 where the Born-Oppenheimer approx-imation breaks down38, 40–42 (for a recent study on geomet-ric phases in polyatomic molecules from a gauge field the-ory perspective, see Ref. 43). In a recent work,44 we havelooked for alternative manifestations of the PVED in isolatedchiral molecules (described as a pure state) by analyzing thegeometric phase as a consequence of the stereomutation dy-namics of these molecules. In particular, we have derived theexplicit dependence of the geometric phase on the essen-tial parameters describing this stereomutation process. Evenmore, the geometric phase has also been explored for superpo-sition states by using a canonical formulation in the descrip-tion of the corresponding dynamics45 by means of a Caldeira-Legget-like Hamiltonian.46 External chiral fields (circularlypolarized electric fields) have been shown to influence thegeometric phase. Furthermore, the interference pattern dis-played by such a superposition of states in the presence ofexternal fields exhibits PVED-locking when the amplitude ofthe field equals the PVED in absolute value. This locking isalso exhibited by the geometric phase.

There are several extensions of the geometric phase con-cept to open quantum systems based on quantum trajec-tories issued from the stochastic Schrödinger equation,47, 48

quantum interferometry,49–53 and the kinematic approach.54, 55

However, as far as we know, there is no definition of the geo-metric phase in the Langevin framework which is equivalentto the usual density matrix approach.46 In this work, a simpledefinition for the geometric phase acquired by chiral states inthe corresponding cyclic evolution when interactions with theenvironment are taken into account is proposed and analyzed.

0021-9606/2012/136(17)/174505/6/$30.00 © 2012 American Institute of Physics136, 174505-1


http://dx.doi.org/10.1063/1.4707735http://dx.doi.org/10.1063/1.4707735http://dx.doi.org/10.1063/1.4707735mailto: [email protected]

174505-2 Dorta-Urra et al. J. Chem. Phys. 136, 174505 (2012)

In particular, the consequences of the competition betweentunneling and parity violation when an Ohmic friction is as-sumed in an eventual measurement of the geometric phaseand the decoherence displayed by interference patterns in thischiral dynamics are studied.

II. THEORY

A. General considerations

Let us remember that an isolated chiral molecule canbe modelled in a phenomenological way by a two-wellpotential within the Born-Oppenheimer approximation. Weconsider that the chiral states, |L〉 and |R〉, have the sameenergy E0. These states are connected to each other by tunnel-ing through the double well barrier. If Ĥ is the Hamiltonianof the molecule then 〈L|Ĥ |R〉 = −δ, with δ > 0, followinga common criteria for signs. When a slight asymmetry in thedouble well potential is included due to the electroweak parityviolation, the PVED is defined as 2� = 〈L|Ĥ |L〉 − 〈R|Ĥ |R〉.Taking E0 = 0 as the origin of energies, the Hamiltonian de-scribing the molecule is

Ĥ = δσ̂x + �σ̂z, (1)where σ̂x,z are the Pauli matrices (and ¯= 1). The chiral statescan be described in terms of the eigenstates by means of arotational angle θ given by tan 2θ = δ/�.

Following Aharonov and Anandan,37 the geometricphase acquired by the pure state |L〉 during a single oscilla-tion of period T in a cyclic evolution is

φg(L) = π +∫ T

0〈L|i d

dt|L〉dt = π +

∫ T0

〈Ĥ 〉|L〉, (2)

which turns out to be44

φg(L) = −φg(R)

= π (1 − cos 2θ ) = π(

1 − �√δ2 + �2

), (3)

and where the second term is known as the dynamical phase.Thus, chiral states only acquire a geometric phase due tostereomutation iff parity is non-conserved. When tunneling isprevented due to a high barrier, the splitting between chiralstates is only due to the PVED and an eventual measurementof the geometric phase should be strictly zero. Let us notethat, as interactions with the environment lead to tunnel sup-pression (see, for example, the dissipative-induced racemiza-tion shown in Ref. 45), a strong dependence of the geometricphase on the PVED is expected when considering either longtimes or strong dissipation, irrespective of the height of thebarrier.

B. Generalized Langevin dynamicsof chiral molecules

An alternative and very appealing way to express the ge-ometrical phase of a linear combination of chiral states is bymeans of a canonical formulation.44, 45 Let us consider a state

of the form

|�(t)〉 = aL(t)|L〉 + aR(t)|R〉. (4)Then, if the corresponding complex amplitudes are written inpolar form as aL,R(t) = |aL,R(t)|eiL,R(t), the optical activityis defined as z(t) ≡ |aR(t)|2 − |aL(t)|2, and the phase differ-ence between chiral states as (t) ≡ R(t) − L(t), then thegeometric phase for this superposition of chiral states is givenby

φg(�) = π + π δ

(−

√1 − z20 cos 0 +

�

δz0

), (5)

where z0 = z(t = 0), 0 = (t = 0), and =√

δ2 + �2.Any isolated two-level system has been showed to be de-

scribed by the Hamiltonian45

H = −√

1 − z2 cos + �δz, (6)

where z and can be seen as a pair of canonically conju-gate variables. In the quantum domain, these variables obeythe Heisenberg equations of motion which are formally iden-tical to the Hamilton equations of motion, ż = −∂H/∂ anḋ = ∂H/∂z. Thus, in terms of these two new conjugate vari-ables, the geometric phase given by Eq. (5) turns out to be44

φg(�) = π(

1 + δ

H0

), (7)

where 〈Ĥ 〉� = H0 ≡ H [z(t = 0),(t = 0)] denotes the av-erage value of the molecular energy which is a conservedmagnitude when dealing with a closed system. It should benoticed that the first term of the Hamiltonian (6) accounts forthe tunneling process and the second one for the underlyingPVED asymmetry.

Noting that and z play the role of a generalized co-ordinate and momentum, respectively, one can introduce in-teractions with the environment by means of a system-bathbilinear coupling via a Caldeira-Legget-like Hamiltonian ex-pressed as44

H = −√

1 − z2 cos + �δz

+ 12

∑i

(�iz

2i +

2i ω2i

�i

)

−∑

i

cii + 2∑

i

c2i �i, (8)

where the sums run over the coordinates of the bath oscil-lators {zi, i} and �i, ci, and ωi are suitable dimensionlessconstants representing generalized masses, couplings with theenvironment, and oscillator frequencies, respectively. Withoutloss of generality, notice that these bath variables could alsobe alternatively written in terms of the usual harmonic oscilla-tor ones (positions and momenta). In any case, these environ-mental degrees of freedom can be exactly eliminated from theequations of motion. On the other hand, when dealing withphases or angles, the bilinear coupling could be questionabledue to the fact that the corresponding Hamiltonian does notdisplay the periodicity of the problem at hand. For example,



in the rotational tunneling process (see Ref. 56 and referencestherein) the environmental coupling has the same periodicityas the hindering potential. In our canonical formalism for chi-ral molecules, the generalized position is a phase variable andthe molecule-bath coupling should be periodic in . Thus, wecould consider the following generalized Hamiltonian:

H = −√

1 − z2 cos + �δz

+ 12

∑i

(�iz

2i +

2i ω2i

�i

)

− f ()∑

i

cii + f 2()∑

i

c2i �i, (9)

where the periodicity of the coupling is given by the specificchoice of the function f(). The corresponding coupled gen-eralized Langevin equations are then given by (dimensionlesstime units are used, t → 2δt)

ż = −√

1 − z2 sin

− df ()d

∫ t0

γ (t − t ′)df ()d

̇(t ′) dt ′ + df ()d

ξ (t),

̇ = z√1 − z2 cos +

�

δ, (10)

where

γ (t) =∑

i

�ic2i cos ωi(t − t ′) (11)

is the damping kernel and

ξ (t) = ∑i ci�i (i(0) cos ωit + zi(0) sin ωit)− c2i �i0f (0)

df ()

dcos ωit (12)

represents a stochastic force, depending on the initial condi-tions of both the system and the bath. The temperature willbe considered high enough in order to neglect contributionscoming from the zero point motions of bath oscillators. Thus,when averaging over the bath variables, this force vanishesleading to a pure dissipative dynamics in terms of the av-erage quantities of the system which will be again denoted(for simplicity) by z and . In other words, Eqs. (10) willbe solved numerically for the average quantities and wherethe average stochastic force is zero. We point out that whenreaching cold temperatures, it would be necessary to consideronly the two lowest states thermally populated and the noiseshould be treated quantum mechanically.

If only dissipative dynamics is considered, Eqs. (10) forthe average quantities, z and , can be derived from the fol-lowing effective Hamiltonian:

Hγ (t) = −√

1 − z2 cos + �δz

+ γ ̇∫

0

(df (′)d′

)2d′, (13)

which explicitly depends on friction and the generalized ve-locity, ̇. Let us point out that Hγ actually represents the non-conserved energy of the chiral system in the presence of dis-sipation due to the coupling with the bath.

At this point, a quite natural and straightforward exten-sion of the geometric phase, displayed by a superposition ofchiral states during a cycle of period T, can be proposed fromEq. (2) to the case of dissipative dynamics as follows:

φγg (�) ≡ π +∫ T

0Hγ (t) dt. (14)

This quantity plays the same role as that used in the defini-tion of geometric phase given by Eq. (2). After Eq. (14), it iseasy to see that the non-dissipative phase is recovered takingγ = 0 and noting that Hγ → H0, which is a conserved magni-tude (we would like to point out the formal similarities be-tween the geometric phase given by our Eq. (14) and thatof Ref. 55 (Eq. 21), derived from Lindblad’s formalism). Itshould also be stressed here the importance of the dissipatedenergy in one period of the cyclic evolution for the evaluationof the geometric phase.

C. Role of the coupling function

Let us now analyze the effect of different generalizedcouplings in the dissipative dynamics. If Ohmic friction is as-sumed, the generalized Langevin equations given by Eqs. (10)are rewritten as

ż = −√

1 − z2 sin − γ(

df ()

d

)2̇,

̇ = z√1 − z2 cos +

�

δ. (15)

If the usual bilinear coupling is considered, the dissipated en-ergy is given by

Hγ (t) = −√

1 − z2 cos + �δz + γ̇. (16)

On the other hand, when periodic coupling is assumed, themost standard functions to be considered are a sine or co-sine function, f (1)() = √2 sin and f (2)() = √2 cos .The following effective Hamiltonians are straightforwardlyobtained to be

H (1,2)γ (t) = −√

1 − z2 cos + �δz

+ γ ̇(

± sin 22

)(17)

where the superscripts (1, 2) correspond to the plus and mi-nus sign, respectively. For the cosine-type coupling, when thephase is a small quantity, the term involving the friction coef-ficient is very small or even zero for a certain period of time.This fact should lead to a lower decaying behavior with time.And, finally, depending on the initial conditions, the startingvalue of the decaying system energy will be different.

D. Decoherence of interference patterns

For this dissipative dynamics, information on interfer-ence experiments can be straightforwardly extracted fromthe probability density. The typical interference signal is ofcosine-type according to

I ∝ |�(t)|2 ∝ 1 + 2|aL(t)||aR(t)| cos (t) (18)



and one should be able to obtain information of the phasefrom the cosine oscillations. Using the effective Hamiltonianapproach here developed, the total intensity of the interfer-ence pattern for the Ohmic case is given now by

I ∝ 1 + �δz − Hγ = 1 +

√1 − z2 cos

− γ ̇∫

0

(df (′)d′

)2d, (19)

where z, are the solutions of Eqs. (15). This type of experi-ments could also be used to look for signals of parity violationin chiral molecules. Furthermore, the presence of dissipationin this effective Hamiltonian will lead to the suppression ofquantum interference, that is, to decoherence. The quantuminterference (coherence) is given by Eq. (19) when replacingHγ by H0.

III. RESULTS

It is very instructive to analyze first how the energy isdissipated with time for different coupling functions when thetunneling or the PVED asymmetry is competing. This magni-tude has been previously shown to play a critical role in theevaluation of the geometric phase and the time evolution ofinterference patterns.

For the bilinear coupling, Fig. 1 shows the time depen-dence of Hγ for � = 0 (symmetric wells) and δ = 0.5 at dif-ferent values of the friction coefficient, γ (given in units of2δ), and with initial conditions z0 = 0.999 and 0 = 0. Inthis case, the tunneling process only rules the dynamics andHγ displays a global exponential relaxation. With the correctdimensions, the expected asymptotic behavior is reproduced,that is, Hγ → −2δ for long times. This asymptotic limit isdue to the tunneling blocking which occurs with friction isacting at very long times. The linear term involved in Eq. (6)plays no role. Therefore, the system tends to an eigenstate ofthe isolated chiral molecule. Figure 1 also shows, for γ = 0,the conserved energy of the system. This behavior is expectedto occur, for example, in molecules such as H2O2 or H2S2,which have a ratio of �/δ 10−15 and 10−6, respectively.7

0 10 20 30 401.0

0.8

0.6

0.4

0.2

0.0

t

H

FIG. 1. Time dependence of Hγ for different values of the friction [γ = 0(dotted), γ = 0.1 (solid), and γ = 0.3 (dashed)] when tunneling only rulesthe dynamics of stereomutation (� = 0, symmetric wells).

0 10 20 30 40

5

0

5

t

H

FIG. 2. Time dependence of Hγ for different values of the PVED [�/δ = 0(solid line), �/δ = 1.2 (dotted line), and �/δ = 5 (dashed line)]. We have takenγ = 0.1, z0 = 0.999, and 0 = 0.

A drastic scenario takes place when the PVED is present,� = 0. As shown in Fig. 2, and for different ratios �/δ, noglobal behavior is observed in the time evolution of the dissi-pated energy, Hγ (t), except for the solid curve which displaysthe behavior with no PVED asymmetry. The other two valuesfor the ratio �/δ = 1.2, 5 correspond to two drastically dif-ferent dynamics. The first value describes a dynamics wherethe tunneling and the PVED asymmetry are competing and,in the second value, the dynamics is without tunneling. Atlong times, in both cases, the system energy tends again tothe eigenstate −2 = −2√δ2 + �2 following a different be-havior. In the competing case, the oscillations displayed bythe dissipative system are due to the presence of tunneling ex-cept to very long times where this process is inhibited due tothe friction. On the contrary, in the second case, the asymp-totic energy is reached after a short period of time but in anabrupt way since tunneling is not present and the linear termof Hamiltonian (6) dominates the dynamics. An inspection ofFig. 2 reveals that the most rich structure occurs for moleculessuch that �/δ 1, as is the case of D2Se2.7

In Fig. 3, the comparison between the dissipative dynam-ics driven by bilinear, sine, and cosine coupling functions is

0 10 20 30 401.0

0.8

0.6

0.4

0.2

0.0

t

H

FIG. 3. Time dependence of Hγ for �/δ = 0, γ = 0.1 and different couplings(bilinear, sine, and cosine dissipations depicted by solid, dotted, and dashedlines, respectively). We have taken z0 = 0.999 and 0 = 0 for the initialconditions.



0 10 20 30 40

10

5

0

5

10

t

H 0 2 4 6 8 10

10

5

0

5

10

t

H

FIG. 4. Time dependence of Hγ for �/δ = 5 and different couplings (solid,dotted, and dashed for bilinear, sine, and cosine coupling functions, respec-tively). We have taken z0 = 0.999 and 0 = 0 as the initial conditions of thedynamics.

shown for the case of parity-conserving molecules (such asH2O2). The dissipative dynamics is shown to be not very dif-ferent in case of bilinear and sine couplings, with the excep-tion of small discrepancies shown regarding the decay time.On the contrary, when a cosine-type coupling is considered,dissipative effects are shown to be strongly reduced. As ex-pected, the elapsed time needed to reach complete localiza-tion is enormously high in the case of cosine coupling.

As observed in Fig. 4, when the PVED asymmetry dom-inates mainly the chiral dynamics, �/δ = 5, the main dif-ferences among the bilinear, sine-, and cosine-type couplingfunctions are clearly manifested at long times. Thus, whenthe dissipative dynamics is due to both bilinear and sine-typecouplings, the energy tends very rapidly to the correspond-ing eigenstate. On the contrary, for cosine dissipation, the en-ergy performs regular oscillations around the same asymp-totic value, which will be reached for very, very long times.

Once the dissipated energy has been analyzed in termsof the different coupling functions, the next step is to showhow the geometric phase depends on the friction coefficient.The explicit dependence of the dynamical phase (when theπ -phase is neglected) on the friction coefficient γ is deducedfrom solving numerically Eqs. (15) and then using the pre-vious definition given by Eq. (14). In Fig. 5, the geometricphase is displayed when � = 0 (solid line) and �/δ = 0.3(solid line with points) for the bilinear coupling. We notethat φγg → π (δ/)H0 when γ → 0 according to Eq. (7). Onthe contrary, with the friction, the geometric phase reachesa plateau given by φg(γ → ∞) → 2π . This behavior is adirect consequence of tunneling suppression. Thus, one canspeculate on using an appropriate tuning of the environment(by changing the pressure, the temperature, or the density ofa surrounding gas, for example) to test the dependence of thegeometric phase on the friction coefficient and on the PVED.Moreover, when a cosine-type coupling is considered (pointsin Fig. 5), noticeable differences can be shown between thegeometric phase in the bilinear case and in the present situa-tion. It should be emphasized that, when the PVED is present,the plateau displayed by the geometric phase is reached be-

0

1

2

3

4

5

6

7

0 2 4 6 8 10

Geo

met

ric P

hase

Friction Coefficient

FIG. 5. Geometric phase in terms of the friction for different values of thePVED [� = 0 (solid line) and �/δ = 0.3 (solid line with points)]. We havetaken z0 = 0.999 and 0 = 0 as the initial conditions of the chiral dynam-ics. The points (without line) represent the geometric phase for a cosine-typecoupling and � = 0.

fore (at slightly smaller friction coefficients) compared to thesituation in which tunneling prevails.

Finally, Fig. 6 displays the time-dependent interferencesignal for different kinds of coupling for γ = 0.1 and whenPVED dominates over the tunneling process. For comparison,we have also depicted the non-dissipative case (dashed-dottedline). When γ = 0, the interference signal displays regular co-sine oscillations for all times (coherence). With Ohmic fric-tion, in all cases, the coherence is destroyed (decoherence). Ifa bilinear coupling is assumed, strong oscillations are presentat short times, although the final damping of the signal can beappreciated when t > 5. The main differences between sine(dotted line) and cosine dissipation (dashed line) appear att > 5. In this region, damping leads to an interference signalwhich is quite similar to the one shown by the bilinear case.On the contrary, as expected, with cosine dissipation, damp-ing gives place to an oscillating signal whose amplitude isagain very slowly decreased. At short times, similar behav-iors are displayed for different coupling functions.

0 2 4 6 8 10

15

10

5

0

5

t

I

FIG. 6. Interference signal for �/δ = 5 and bilinear (solid line), sine- (dottedline), and cosine-type couplings (dashed line). The non-dissipative signal isdepicted with dashed-dotted lines. We have taken γ = 0.1, z0 = 0.999, and0 = 0.



IV. CONCLUSION

In this work, we have put in evidence the fundamentalrole played by the parity-violating energy difference in thedissipative stereomutation dynamics, within a canonical de-scription of chiral molecules in the Langevin framework. Thenonlinearity exhibited by the coupled equations describingthis process competing with tunneling has shown to be veryrich. The canonical formalism has allowed us to define aneffective Hamiltonian for Ohmic friction for any dissipativetwo-level system interacting with a bath of harmonic oscilla-tors via a generalized coupling. In particular, the effects of twodifferent periodic (sine and cosine) coupling functions havebeen rationalized in terms of strong and weak dissipation, re-spectively. A straightforward extension of the definition of thegeometric phase in pure states to open quantum systems withOhmic friction has been proposed. Finally, the role of differ-ent dissipative effects in the energy, the geometric phase, anddecoherence displayed by interference patterns of this cyclicevolution has been discussed.

ACKNOWLEDGMENTS

This work has been funded by the MICINN (Spain)through Grant Nos. CTQ2008-02578, FIS2010-18132, andby the Comunidad Autónoma de Madrid, Grant No.S-2009/MAT/1467. P.B. acknowledges a Juan de la Ciervafellowship from the MICINN and A.D.-U. acknowledges aJAE fellowship from CSIC. H.C.P.-R. and G.R.-L. acknowl-edge a scientific project from INSTEC.

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