- Email: [email protected]
Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates
Computational Condensed Matter xxx (2016) 1e5
Contents lists available at ScienceDirect
Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp
Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates Xian-Ting Liang Department of Physics and Institute of Optics, Ningbo University, Ningbo 315211, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 June 2016 Received in revised form 5 September 2016 Accepted 27 September 2016 Available online xxx
In this paper, we investigate the influences of different environmental factors to rephasing spectra of 2D, third-order, non-linear (2D3ONL), photon echo signals for bio-molecular aggregates by using an exact numerical dynamics approachdthe quasi-adiabatic propagator path integral (QUAPI) method. It is shown that the increase of temperature and coupling strength between the system and its environment results in not only the inhomogeneous but also the homogeneous broadening of the 2D3ONL rephasing spectra of photon echo signals. However, it is interested that the increase of dynamical memory times, namely, the increase of the non-Markovian effects will mainly lead to the homogeneous broadening of the spectra. The evolutions of the 2D3ONL rephasing spectra with population times are also investigated. © 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Two-dimensional (2D) non-linear spectroscopy is an important tool to study the vibrations and electronic excitations of molecular systems. It can be used to investigate the structure and dynamics of complex systems. Numerous studies have involved 2D non-linear spectroscopy on various types of molecular aggregates such as Jaggregates [1] and light harvesting complexes in photosynthetic systems [2], and some important information on the structure, coupling, and energy transfer of these systems has been obtained [3,4]. Therefore, it is important to carefully investigate the related factors of influencing the 2D non-linear spectroscopy. However, there are many factors affecting the pattern of the 2D non-linear spectra. These factors are always described with the model parameters, and some of them can be controlled in experiments, and some others are intrinsic properties of the system and its environments. In photon echo experiments, the influences to the 2D spectroscopy are divided into two kinds, one of which results in the homogeneous broadening, and the other results in the inhomogeneous broadening. In pigment-protein complexes, the slow motions of the protein environment result in a static site-energy distribution, which is called inhomogeneous broadening or static disorder. On the other hand, the system dynamics influenced by the amplitude of the dynamical fluctuations, is known as homogeneous broadening,
E-mail address: [email protected].
which essentially is induced by the coupling of the system to the vibrational environment. However, in general it is not one to one correspondence between the model parameters and types of spectral broadening. In the 2D non-linear spectroscopy, the broadening of the signal along the diagonal line (u1 ¼ u3) refers to the homogeneous broadening, and the elongation along the diagonal line denotes the inhomogeneous broadening according to Ref. [5]. It is interested that what parameters lead to the inhomogeneous broadening and what parameters to the homogeneous broadening. In this paper, we will show that the increase of the temperature and coupling strength between the system and its environment results in not only the inhomogeneous but also homogeneous broadening. However, the increase of the nonMarkovian effects will mainly lead to the homogeneous broadening of the 2D3ONL rephasing spectra of the photon echo signals. This observation may be useful to experimentally probe the environments of bio-molecular aggregates. 2. 2D3ONL photon echo spectroscopy 2.1. Model and dynamics method The three-pulse photon echo experiment has been extensively developed and applied [4,6e9]. In this kind of experiments, the system to be studied is probed by three incident laser pulses [10e13]. The first pulse creates a superposition state between the ground state and excited state, which evolves for time delays t1 until the second pulse interacts with the sample. The second pulse
http://dx.doi.org/10.1016/j.cocom.2016.09.006 2352-2143/© 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
2
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
creates a population in the exciton system, and the corresponding dynamics is recorded by the population time t2. The third pulse interacts with the system to generate a photon echo signal, and t3 stands for the signal time. By carefully matching the phases and directions of the input laser pulses, one can obtain the output of the echo signals correlated with the first and third pulses. Normally, the rephasing and nonrephasing signals from the photon echo are summed, and this sum is phased to acquire corrected rephasing and nonrephasing spectra. However, by employing the Gradient Assisted Photon Echo Spectroscopy (GAPES) technique [11], the rephasing and nonrephasing signals from pump-probe data can be phased independently. Thus, the experimental rephasing and nonrephasing spectra can be compared with the calculated ones. ! ! Here, three laser pulses projected onto the sample along k 1 , k 2 , ! and k 3 , and the detected third-order non-linear response signals ! along the phase-matched directions k s . In the impulsive limit, the rephasing and nonrephasing spectra are calculated as the real part of the following double Fourier-Laplace transforms of Rr(t1,t2,t3) and Rnr(t1,t2,t3) with respect to t1 and t3, respectively:
Z∞
Z∞ Ir;nr ¼
dt1 0
dt3 eiðHu1 t1 þu3 t3 Þ Rr;nr ðt1 ; t2 ; t3 Þ:
Hsb
2 !2 3 N X p2 X 1 ci~sjnihnj 5 2 i 4 ; ¼ þ mu x 2mi 2 i i n¼0 i mi u2i i
(4)
in Eq. (3). Here, xi, pi, mi, and ui are the coordinates, momenta, masses, and frequencies of the bath modes, respectively; ci are the coupling coefficients between the modes of the system and the bath; and jn〉ðn ¼ 0; 1; 2Þ are the basis states corresponding to the two-site system (where j0〉 is the ground state, and the double excited state is not considered). If the interaction between the system and the bath is switched on at t ¼ 0, i.e., if the initial density matrix has the product form r(0) ¼ rs(0)rb(0), the evolution of the reduced density matrix of the system,
Ei 00 hD r~ s0 s ; t ≡Trb s00 eiHt=Z rð0ÞeiHt=Z s0 ;
(5)
is given by the quasi-adiabatic propagator path integral expression
(1)
0
! ! Here, Rr and Rnr(t1,t2,t3) are the rephasing ( k s ¼ k 1 þ ! ! ! ! ! ! k 2 þ k 3 ) and nonrephasing ( k s ¼ k 1 k 2 þ k 3 ) signals, which are detected in experiments and can be calculated theoretically through [14e16].
n o eq Rr;nr ðt1 ; t2 ; t3 Þ ¼ Tr m Gðt3 Þm þ Gðt2 Þm± Gðt1 ÞmH rtot ð0Þ :
(QUAPI) method developed by Makri's group [22,23,32e34]dto simulate the 2D3ONL rephasing spectra of the photon echo signals for the system. In this method we need not use the rotating wave approximation of the Hamiltonian; namely, we set
(2)
Here, G(t) is the retarded propagator of the total system in P , and m m m x ¼ ½ ; x, where ≡ Liouville space, m mih0 ± þ m0 ± m P m ≡ m0m 0ihm, corresponding to dipole operator m ¼ mþþm, m
Z Z Z Z Z 00 Z þ þ ds ds ds r~ s ; s0 ; t ¼ dsþ / ds / ds 0 1 N1 0 1 N1 00 þ iHs DtZ þ s eiHs Dt=Z sþ s0 N1 / s1 e sþ 0 rs ð0Þs0 iH Dt=Z e s s / s eiHs Dt=Z s0 s 0 N1 1 þ þ 00 0 I sþ 0 ; s1 ; …; sN1 ; s ; s0 ; s1 ; …; sN1 ; s ; Dt : (6) It can be shown that the propagator tensor is decided not only by the bare system propagator matrix,
and we suppose the total system to be the equilibrium state, initially, represented with req tot ð0Þ. t1 ¼ t2 t1 , t2 ¼ t3 t2 , and t3 ¼ t t3 , where t1 ; t2 , and t3 are the time points of the laser projecting onto a sample, and t is the time point of receiving echo signals. In the following we investigate the rephasing spectrum by using a two-site model. The Hamiltonian of the open system is
ED E D ± þ iHs Dt=Z þ iHs Dt=Z ¼ s s K s± ; s e s e sk ; kþ1 kþ1 k k kþ1
H ¼ Hs þ Hsb þ Hb ;
0 where sþ N ¼ s , and sN ¼ s . The coefficients hkk are given in Ref. [32]. Numerically, the influence functional is given by
(3)
where Hs and Hb are the Hamiltonians of the two-site system and its bath, and Hsb is the interaction Hamiltonian between the system and its bath. Several kinds of theoretical methods have previously been developed to calculate the 2D spectra through this kind of model. The cumulant expansion method based on the standard perturbation theory was first introduced by Mukamel and coworkers [8]. Later, the master equations of Redfield form [17] and modified Redfield form [18] were introduced for this kinds of investigations. Recently, non-Markovian exact numerical methods, the hierarchical equation of motion method [16,19e21], the numerical path integral method [22e24], the non perturbative method [15], and other methods [25,26] were used to investigate the non-linear spectra. For the open systems with continuous variables, the response functions and the 2D non-linear spectra have also been investigated by using analytical and numerical methods [27e31]. In this paper, we use an exact numerical dynamics approachdthe quasi-adiabatic propagator path integral
(7)
but also by an influence functional,
# N X k 1X þ þ sk sk hkk0 sk0 hkk0 sk0 : I ¼ exp Z 0 "
(8)
k¼0 k ¼0
00
± ± I s± n ; snþ1 ; …; snþDk
max
± ± ¼ I0 s± n I1 sn ; snþ1 ± ± ± I2 s± n ; snþ2 /IDkmax sn ; snþDk
max
: (9)
Here,
± ± Iðs± n ; snþ1 ; …; snþDkmax Þ
describes the memory effects of the
system. It is clear that, in the memory time, the memory effects are included much more when Dkmax is larger. The length of the memory time can be estimated by the following bath response function
a¼
1
p
Z∞
bZu cosut isinut : duJðuÞ coth 2
(10)
0
It shows that there are different memory times when the same
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
3
Fig. 1. Real part of the response functions for the Ohmic bath at different cut-off frequencies. Here, we set the temperature T ¼ 100 K, and x ¼ 0:5. The correlation times are about 60 fsas uc ¼ 1700 cm1(a), 120 fsas uc ¼ 1400 cm1(b), and 180 fsas uc ¼ 1000 cm1(c).
system in different environments which have different cut-off frequencies [32,35]. In Fig. 1 we plot the real part of the response functions when the baths are described with Ohmic spectral density functions of different cut-off frequencies uc ¼ 1700 cm1 (a), 1400 cm1 (b), and 1000 cm1 (c). It can be easily seen that their memory times are about 60,120, and 180 fs in above three cases. Here, we set temperature T ¼ 100 K, and x ¼ 0:5 Thus, if we fix the cut-off frequency of the bath, the dynamics calculations will include different non-Markovian effects as taking different correlation times [36e38]. By using the QUAPI method and Eqs. (1) and (2), we can simulate the 2D3ONL rephasing spectra of the photon echo signals for the open two-site system. Here, the system Hamiltonian is set as Hs ¼ ½50j1〉〈1j þ 100j2〉〈2j 15j2〉〈1j 15j1〉〈2j cm1, and the strength of the dipole is set as m01 ¼ m10 ¼ m02 ¼ m20 ¼ 1. It is should be noted that this is not the spin-boson model, and here three levels are included, and the three basis states are denoted by j0〉; j1〉 and j2〉. We suppose that the energy of the ground state is zero, and there is no coupling between the ground state with other states in the Hamiltonian. However, the laser pulses can result in dipole transition between the ground state and the excited states. We use the ohmic spectral density function to model the environment: JðuÞ ¼ xueu=uc . 2.2. Results and discussions In the following, we investigate the broadening of the 2D3ONL rephasing spectra of photon echo signals as considering different parameter conditions by using the QUAPI dynamics method [23]. Here, we set the time step Dt ¼ 30 fs. It has been checked that the calculations are converged in this time step. At first, we plot the 2D3ONL rephasing spectra of photon echo signals for the system in different temperature T ¼ 100 K, 200 K, and 300 K, as other parameters are fixed. Here, we choose the cutoff frequencies of the bath uc ¼ 1400 cm1 s, population time
t2 ¼ 300 fs, and Dkmax ¼ 3. It is shown that the 2D3ONL rephasing spectra of the photon echo signals are broadened with the increase of the temperature. It is clear from Fig. 2 that when the temperature increase there are not only the inhomogeneous but also the homogeneous broadening. Secondly, we investigate the 2D3ONL rephasing spectra of the photon echo signals for the system in different coupling coefficient x ¼ 0:2; 0:4, and 0.6, when other parameters are fixed as them in Fig. 3, and the temperature is set as T ¼ 100 K. It is shown that in these cases there are both the homogeneous and inhomogeneous broadening, but the homogeneous broadening is much more than the inhomogeneous broadening, see Fig. 3. Thirdly, we investigate the 2D3ONL rephasing spectra of the photon echo signals for the system in different cut-off frequencies of the bath corresponding to three kind of cases plotted in Fig. 1, as other parameters are fixed as T ¼ 100 K, and x ¼ 0:5. Fig. 4 shows the broadening of the 2D3ONL rephasing spectra when we choose three different the cut-off frequencies as uc ¼ 1700 cm1, 1400 cm1, and 1000 cm1. As plotted in Fig. 1, the memory times are different in the three cases. In order to include the major nonMarkovian effects, we choose Dkmax ¼ 2, 4, and 6 (see Fig. 4 (a), (b), and (c), respectively). It is clearly shown that the homogeneous broadening are much obvious than the inhomogeneous broadening in the 2D3ONL rephasing spectra when only the memory time increase. This observations may be use to distinguish different environments. Finally, we investigate the 2D3ONL rephasing spectra of the photon echo signals for the system in different population time t2, as the parameters h ¼ 0.5, uc ¼ 1400 cm1, temperature T ¼ 100 K, and Dkmax ¼ 4. As expected that the signals decays with the increase of the population time, see Fig. 5. 3. Conclusions In this paper, we investigate the broadening of the 2D3ONL
Fig. 2. Rephasing spectra of a two-site system in different temperature T ¼ 100 K (a), 200 K (b), and 300 K (c), where the numerical step is set as Dt ¼ 30 fs, and t2 ¼ 300 fs. Here, we choose the bath spectral density function as the Ohmic form with x ¼ 0:5, and uc ¼ 1400 cm1, Dkmax ¼ 3.
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
4
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
Fig. 3. Rephasing spectra of a two-site system in different coupling coefficients x ¼ 0:4 (a), 0.5 (b), and 0.6 (c), where the temperature is chosen as T ¼ 100 K. Other parameters are the same as them in Fig. 2.
Fig. 4. Rephasing spectra of a two-site system in Dkmax ¼ 2 (a), 4 (b), and 6 (c), which correspond to uc ¼ 1700 cm1, 1400 cm1, and 1000 cm1, respectively. Here, x ¼ 0:5, and the values of other parameters are the same as them in Fig. 3.
rephasing spectra of bio-molecular aggregates in photon echo experiments. In this investigation we use an exact numerical dynamics approachdthe quasi-adiabatic propagator path integral (QUAPI) method to calculate the response function and then the
rephasing spectra of photon echo signals. This dynamics method is non-Markovian and the more non-Markovian effects are included when longer memory times are considered. It is shown that the increase of the temperature and coupling strength between the
Fig. 5. Rephasing spectra of a two-site system in t2 ¼ 30 (a), 300 (b), 900 (c), 2250 (d), 3750 (e), 5400 (f) fs, where x ¼ 0:5, uc ¼ 1400 cm1, temperature T ¼ 100 K, and Dkmax ¼ 4.
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
system and its environment results in both the inhomogeneous and homogeneous broadening. However, the increase of the memory times will mainly lead to the homogeneous broadening of the 2D3ONL rephasing spectra of the photon echo signals, which means that the non-Markovian effects mainly results from the couplings of the system and the modes of its environments. The rephasing signals of photon echo decay with the population time increase. These observations may help us to exactly probe the biomolecular aggregate, its environment, and the coupling between them. Acknowledgments This project was sponsored by the the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY13A040006), and the K. C. Wong Magna Foundation in Ningbo University.
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
References [1] I. Stiopkin, Tobias Brixner, Mino Yang, G.R. Fleming, J. Phys. Chem. B 110 (2006) 20032. [2] E. Collini, C.Y. Wong, K.E. Wilk, P.M.G. Curmi, P. Brumer, G.D. Scholes, Nature 463 (2010) 644. [3] P. Huo, D.F. Coker, J. Phys. Chem. Lett. 2 (2011) 825. [4] T. Brixner, J. Stenger, H.M. Vaswani, M. Cho, R.E. Blankenship, G.R. Fleming, Nature 434 (2005) 625. [5] M. Cho, T. Brixner, I. Stiopkin, H. Vaswani, G.R. Fleming, J. Chin. Chem. Soc. 53 (2006) 15e24. [6] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, vol. 29, Oxford University Press New York,, 1995. [7] P. Hamm, M. Zanni, Concepts and Methods of 2D Infrared Spectroscopy,
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
5
Cambridge University Press, 2011. S. Mukamel, Annu. Rev. Phys. Chem. 51 (2000) 691. Y.-C. Cheng, R.E. Blankenship, G.R. Fleming, Nature 446 (2007) 12. W.G. Noid, R.F. Loring, Mol. Phys. 103 (2005) 3071. V. Singh, A. Fidler, B. Rolczynski, G. Engel, J. Chem. Phys. 139 (084201) (2013). S. Woutersen, P. Hamm, J. Phys. Chem. B 104 (2000) 11316. S. Park, K. Kwak, M. Fayer, Laser Phys. Lett. 4 (2007) 704. A. Ishizaki, Y. Tanimura, Chem. Phys. 347 (2008) 185. L. Seidner, G. Stock, W. Domcke, J. Chem. Phys. 103 (1995) 3998. L. Chen, R. Zheng, Q. Shi, Y. Yan, J. Chem. Phys. 132 (2010) 227. A.V. Pisliakov, M. Toms, G.R. Fleming, J. Chem. Phys. 124 (2006) 7749. W.M. Zhang, T. Meier, V. Chernyak, S. Mukamel, J. Chem. Phys. 108 (1998) 7763. A. Ishizaki, Y. Tanimura, J. Phys. Soc. Jpn. 74 (2005) 3131. A. Ishizaki, Y. Tanimura, J. Chem. Phys. 125 (2006) 10369. I. Akihito, T. Yoshitaka, J. Phys. Chem. A 111 (2007) 9269. M.M. Sahrapour, N. Makri, J. Chem. Phys. 132 (2010) 134506. X.-T. Liang, J. Chem. Phys. 141 (044116) (2014). X.-T. Liang, Chin. Phys. Lett. 29 (084211) (2012). C. Minhaeng, H.M. Vaswani, B. Tobias, S. Jens, G.R. Fleming, J. Phys. Chem. B 109 (2005) 10542. H.G. Duan, A.G. Dijkstra, P. Nalbach, M. Thorwart, Phys. Rev. E 92 (042708) (2015). Y. Tanimura, S. Mukamel, J. Chem. Phys. 99 (1993) 9496. Y. Tanimura, Y. Maruyama, J. Chem. Phys. 107 (1997) 1779. J. Wu, J. Cao, J. Chem. Phys. 115 (2001) 5381. W.G. Noid, G.S. Ezra, R.F. Loring, J. Chem. Phys. 120 (2004) 1491. W.G. Noid, G.S. Ezra, R.F. Loring, J. Phys. Chem. B 108 (2004) 6536. N. Makri, J. Math. Phys. 36 (1995) 2430. E. Sim, N. Makri, J. Phys. Chem. B 101 (1997) 5446. E. Sim, N. Makri, Comput. Phys. Commun. 99 (1997) 335. X.-T. Liang, Phys. Rev. B 72 (2005) 245328. M. Ali, Phys. Lett. A 378 (2014) 2048. F. Benatti, F. Carollo, R. Floreanini, H. Narnhofer, Phys. Lett. A 380 (2016) 381. J. Piilo, K. Harkonen, S. Maniscalco, K.A. Suominen, Phys. Rev. A 79 (062112) (2009).
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
Contents lists available at ScienceDirect
Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp
Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates Xian-Ting Liang Department of Physics and Institute of Optics, Ningbo University, Ningbo 315211, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 June 2016 Received in revised form 5 September 2016 Accepted 27 September 2016 Available online xxx
In this paper, we investigate the influences of different environmental factors to rephasing spectra of 2D, third-order, non-linear (2D3ONL), photon echo signals for bio-molecular aggregates by using an exact numerical dynamics approachdthe quasi-adiabatic propagator path integral (QUAPI) method. It is shown that the increase of temperature and coupling strength between the system and its environment results in not only the inhomogeneous but also the homogeneous broadening of the 2D3ONL rephasing spectra of photon echo signals. However, it is interested that the increase of dynamical memory times, namely, the increase of the non-Markovian effects will mainly lead to the homogeneous broadening of the spectra. The evolutions of the 2D3ONL rephasing spectra with population times are also investigated. © 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Two-dimensional (2D) non-linear spectroscopy is an important tool to study the vibrations and electronic excitations of molecular systems. It can be used to investigate the structure and dynamics of complex systems. Numerous studies have involved 2D non-linear spectroscopy on various types of molecular aggregates such as Jaggregates [1] and light harvesting complexes in photosynthetic systems [2], and some important information on the structure, coupling, and energy transfer of these systems has been obtained [3,4]. Therefore, it is important to carefully investigate the related factors of influencing the 2D non-linear spectroscopy. However, there are many factors affecting the pattern of the 2D non-linear spectra. These factors are always described with the model parameters, and some of them can be controlled in experiments, and some others are intrinsic properties of the system and its environments. In photon echo experiments, the influences to the 2D spectroscopy are divided into two kinds, one of which results in the homogeneous broadening, and the other results in the inhomogeneous broadening. In pigment-protein complexes, the slow motions of the protein environment result in a static site-energy distribution, which is called inhomogeneous broadening or static disorder. On the other hand, the system dynamics influenced by the amplitude of the dynamical fluctuations, is known as homogeneous broadening,
E-mail address: [email protected].
which essentially is induced by the coupling of the system to the vibrational environment. However, in general it is not one to one correspondence between the model parameters and types of spectral broadening. In the 2D non-linear spectroscopy, the broadening of the signal along the diagonal line (u1 ¼ u3) refers to the homogeneous broadening, and the elongation along the diagonal line denotes the inhomogeneous broadening according to Ref. [5]. It is interested that what parameters lead to the inhomogeneous broadening and what parameters to the homogeneous broadening. In this paper, we will show that the increase of the temperature and coupling strength between the system and its environment results in not only the inhomogeneous but also homogeneous broadening. However, the increase of the nonMarkovian effects will mainly lead to the homogeneous broadening of the 2D3ONL rephasing spectra of the photon echo signals. This observation may be useful to experimentally probe the environments of bio-molecular aggregates. 2. 2D3ONL photon echo spectroscopy 2.1. Model and dynamics method The three-pulse photon echo experiment has been extensively developed and applied [4,6e9]. In this kind of experiments, the system to be studied is probed by three incident laser pulses [10e13]. The first pulse creates a superposition state between the ground state and excited state, which evolves for time delays t1 until the second pulse interacts with the sample. The second pulse
http://dx.doi.org/10.1016/j.cocom.2016.09.006 2352-2143/© 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
2
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
creates a population in the exciton system, and the corresponding dynamics is recorded by the population time t2. The third pulse interacts with the system to generate a photon echo signal, and t3 stands for the signal time. By carefully matching the phases and directions of the input laser pulses, one can obtain the output of the echo signals correlated with the first and third pulses. Normally, the rephasing and nonrephasing signals from the photon echo are summed, and this sum is phased to acquire corrected rephasing and nonrephasing spectra. However, by employing the Gradient Assisted Photon Echo Spectroscopy (GAPES) technique [11], the rephasing and nonrephasing signals from pump-probe data can be phased independently. Thus, the experimental rephasing and nonrephasing spectra can be compared with the calculated ones. ! ! Here, three laser pulses projected onto the sample along k 1 , k 2 , ! and k 3 , and the detected third-order non-linear response signals ! along the phase-matched directions k s . In the impulsive limit, the rephasing and nonrephasing spectra are calculated as the real part of the following double Fourier-Laplace transforms of Rr(t1,t2,t3) and Rnr(t1,t2,t3) with respect to t1 and t3, respectively:
Z∞
Z∞ Ir;nr ¼
dt1 0
dt3 eiðHu1 t1 þu3 t3 Þ Rr;nr ðt1 ; t2 ; t3 Þ:
Hsb
2 !2 3 N X p2 X 1 ci~sjnihnj 5 2 i 4 ; ¼ þ mu x 2mi 2 i i n¼0 i mi u2i i
(4)
in Eq. (3). Here, xi, pi, mi, and ui are the coordinates, momenta, masses, and frequencies of the bath modes, respectively; ci are the coupling coefficients between the modes of the system and the bath; and jn〉ðn ¼ 0; 1; 2Þ are the basis states corresponding to the two-site system (where j0〉 is the ground state, and the double excited state is not considered). If the interaction between the system and the bath is switched on at t ¼ 0, i.e., if the initial density matrix has the product form r(0) ¼ rs(0)rb(0), the evolution of the reduced density matrix of the system,
Ei 00 hD r~ s0 s ; t ≡Trb s00 eiHt=Z rð0ÞeiHt=Z s0 ;
(5)
is given by the quasi-adiabatic propagator path integral expression
(1)
0
! ! Here, Rr and Rnr(t1,t2,t3) are the rephasing ( k s ¼ k 1 þ ! ! ! ! ! ! k 2 þ k 3 ) and nonrephasing ( k s ¼ k 1 k 2 þ k 3 ) signals, which are detected in experiments and can be calculated theoretically through [14e16].
n o eq Rr;nr ðt1 ; t2 ; t3 Þ ¼ Tr m Gðt3 Þm þ Gðt2 Þm± Gðt1 ÞmH rtot ð0Þ :
(QUAPI) method developed by Makri's group [22,23,32e34]dto simulate the 2D3ONL rephasing spectra of the photon echo signals for the system. In this method we need not use the rotating wave approximation of the Hamiltonian; namely, we set
(2)
Here, G(t) is the retarded propagator of the total system in P , and m m m x ¼ ½ ; x, where ≡ Liouville space, m mih0 ± þ m0 ± m P m ≡ m0m 0ihm, corresponding to dipole operator m ¼ mþþm, m
Z Z Z Z Z 00 Z þ þ ds ds ds r~ s ; s0 ; t ¼ dsþ / ds / ds 0 1 N1 0 1 N1 00 þ iHs DtZ þ s eiHs Dt=Z sþ s0 N1 / s1 e sþ 0 rs ð0Þs0 iH Dt=Z e s s / s eiHs Dt=Z s0 s 0 N1 1 þ þ 00 0 I sþ 0 ; s1 ; …; sN1 ; s ; s0 ; s1 ; …; sN1 ; s ; Dt : (6) It can be shown that the propagator tensor is decided not only by the bare system propagator matrix,
and we suppose the total system to be the equilibrium state, initially, represented with req tot ð0Þ. t1 ¼ t2 t1 , t2 ¼ t3 t2 , and t3 ¼ t t3 , where t1 ; t2 , and t3 are the time points of the laser projecting onto a sample, and t is the time point of receiving echo signals. In the following we investigate the rephasing spectrum by using a two-site model. The Hamiltonian of the open system is
ED E D ± þ iHs Dt=Z þ iHs Dt=Z ¼ s s K s± ; s e s e sk ; kþ1 kþ1 k k kþ1
H ¼ Hs þ Hsb þ Hb ;
0 where sþ N ¼ s , and sN ¼ s . The coefficients hkk are given in Ref. [32]. Numerically, the influence functional is given by
(3)
where Hs and Hb are the Hamiltonians of the two-site system and its bath, and Hsb is the interaction Hamiltonian between the system and its bath. Several kinds of theoretical methods have previously been developed to calculate the 2D spectra through this kind of model. The cumulant expansion method based on the standard perturbation theory was first introduced by Mukamel and coworkers [8]. Later, the master equations of Redfield form [17] and modified Redfield form [18] were introduced for this kinds of investigations. Recently, non-Markovian exact numerical methods, the hierarchical equation of motion method [16,19e21], the numerical path integral method [22e24], the non perturbative method [15], and other methods [25,26] were used to investigate the non-linear spectra. For the open systems with continuous variables, the response functions and the 2D non-linear spectra have also been investigated by using analytical and numerical methods [27e31]. In this paper, we use an exact numerical dynamics approachdthe quasi-adiabatic propagator path integral
(7)
but also by an influence functional,
# N X k 1X þ þ sk sk hkk0 sk0 hkk0 sk0 : I ¼ exp Z 0 "
(8)
k¼0 k ¼0
00
± ± I s± n ; snþ1 ; …; snþDk
max
± ± ¼ I0 s± n I1 sn ; snþ1 ± ± ± I2 s± n ; snþ2 /IDkmax sn ; snþDk
max
: (9)
Here,
± ± Iðs± n ; snþ1 ; …; snþDkmax Þ
describes the memory effects of the
system. It is clear that, in the memory time, the memory effects are included much more when Dkmax is larger. The length of the memory time can be estimated by the following bath response function
a¼
1
p
Z∞
bZu cosut isinut : duJðuÞ coth 2
(10)
0
It shows that there are different memory times when the same
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
3
Fig. 1. Real part of the response functions for the Ohmic bath at different cut-off frequencies. Here, we set the temperature T ¼ 100 K, and x ¼ 0:5. The correlation times are about 60 fsas uc ¼ 1700 cm1(a), 120 fsas uc ¼ 1400 cm1(b), and 180 fsas uc ¼ 1000 cm1(c).
system in different environments which have different cut-off frequencies [32,35]. In Fig. 1 we plot the real part of the response functions when the baths are described with Ohmic spectral density functions of different cut-off frequencies uc ¼ 1700 cm1 (a), 1400 cm1 (b), and 1000 cm1 (c). It can be easily seen that their memory times are about 60,120, and 180 fs in above three cases. Here, we set temperature T ¼ 100 K, and x ¼ 0:5 Thus, if we fix the cut-off frequency of the bath, the dynamics calculations will include different non-Markovian effects as taking different correlation times [36e38]. By using the QUAPI method and Eqs. (1) and (2), we can simulate the 2D3ONL rephasing spectra of the photon echo signals for the open two-site system. Here, the system Hamiltonian is set as Hs ¼ ½50j1〉〈1j þ 100j2〉〈2j 15j2〉〈1j 15j1〉〈2j cm1, and the strength of the dipole is set as m01 ¼ m10 ¼ m02 ¼ m20 ¼ 1. It is should be noted that this is not the spin-boson model, and here three levels are included, and the three basis states are denoted by j0〉; j1〉 and j2〉. We suppose that the energy of the ground state is zero, and there is no coupling between the ground state with other states in the Hamiltonian. However, the laser pulses can result in dipole transition between the ground state and the excited states. We use the ohmic spectral density function to model the environment: JðuÞ ¼ xueu=uc . 2.2. Results and discussions In the following, we investigate the broadening of the 2D3ONL rephasing spectra of photon echo signals as considering different parameter conditions by using the QUAPI dynamics method [23]. Here, we set the time step Dt ¼ 30 fs. It has been checked that the calculations are converged in this time step. At first, we plot the 2D3ONL rephasing spectra of photon echo signals for the system in different temperature T ¼ 100 K, 200 K, and 300 K, as other parameters are fixed. Here, we choose the cutoff frequencies of the bath uc ¼ 1400 cm1 s, population time
t2 ¼ 300 fs, and Dkmax ¼ 3. It is shown that the 2D3ONL rephasing spectra of the photon echo signals are broadened with the increase of the temperature. It is clear from Fig. 2 that when the temperature increase there are not only the inhomogeneous but also the homogeneous broadening. Secondly, we investigate the 2D3ONL rephasing spectra of the photon echo signals for the system in different coupling coefficient x ¼ 0:2; 0:4, and 0.6, when other parameters are fixed as them in Fig. 3, and the temperature is set as T ¼ 100 K. It is shown that in these cases there are both the homogeneous and inhomogeneous broadening, but the homogeneous broadening is much more than the inhomogeneous broadening, see Fig. 3. Thirdly, we investigate the 2D3ONL rephasing spectra of the photon echo signals for the system in different cut-off frequencies of the bath corresponding to three kind of cases plotted in Fig. 1, as other parameters are fixed as T ¼ 100 K, and x ¼ 0:5. Fig. 4 shows the broadening of the 2D3ONL rephasing spectra when we choose three different the cut-off frequencies as uc ¼ 1700 cm1, 1400 cm1, and 1000 cm1. As plotted in Fig. 1, the memory times are different in the three cases. In order to include the major nonMarkovian effects, we choose Dkmax ¼ 2, 4, and 6 (see Fig. 4 (a), (b), and (c), respectively). It is clearly shown that the homogeneous broadening are much obvious than the inhomogeneous broadening in the 2D3ONL rephasing spectra when only the memory time increase. This observations may be use to distinguish different environments. Finally, we investigate the 2D3ONL rephasing spectra of the photon echo signals for the system in different population time t2, as the parameters h ¼ 0.5, uc ¼ 1400 cm1, temperature T ¼ 100 K, and Dkmax ¼ 4. As expected that the signals decays with the increase of the population time, see Fig. 5. 3. Conclusions In this paper, we investigate the broadening of the 2D3ONL
Fig. 2. Rephasing spectra of a two-site system in different temperature T ¼ 100 K (a), 200 K (b), and 300 K (c), where the numerical step is set as Dt ¼ 30 fs, and t2 ¼ 300 fs. Here, we choose the bath spectral density function as the Ohmic form with x ¼ 0:5, and uc ¼ 1400 cm1, Dkmax ¼ 3.
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
4
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
Fig. 3. Rephasing spectra of a two-site system in different coupling coefficients x ¼ 0:4 (a), 0.5 (b), and 0.6 (c), where the temperature is chosen as T ¼ 100 K. Other parameters are the same as them in Fig. 2.
Fig. 4. Rephasing spectra of a two-site system in Dkmax ¼ 2 (a), 4 (b), and 6 (c), which correspond to uc ¼ 1700 cm1, 1400 cm1, and 1000 cm1, respectively. Here, x ¼ 0:5, and the values of other parameters are the same as them in Fig. 3.
rephasing spectra of bio-molecular aggregates in photon echo experiments. In this investigation we use an exact numerical dynamics approachdthe quasi-adiabatic propagator path integral (QUAPI) method to calculate the response function and then the
rephasing spectra of photon echo signals. This dynamics method is non-Markovian and the more non-Markovian effects are included when longer memory times are considered. It is shown that the increase of the temperature and coupling strength between the
Fig. 5. Rephasing spectra of a two-site system in t2 ¼ 30 (a), 300 (b), 900 (c), 2250 (d), 3750 (e), 5400 (f) fs, where x ¼ 0:5, uc ¼ 1400 cm1, temperature T ¼ 100 K, and Dkmax ¼ 4.
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006
X.-T. Liang / Computational Condensed Matter xxx (2016) 1e5
system and its environment results in both the inhomogeneous and homogeneous broadening. However, the increase of the memory times will mainly lead to the homogeneous broadening of the 2D3ONL rephasing spectra of the photon echo signals, which means that the non-Markovian effects mainly results from the couplings of the system and the modes of its environments. The rephasing signals of photon echo decay with the population time increase. These observations may help us to exactly probe the biomolecular aggregate, its environment, and the coupling between them. Acknowledgments This project was sponsored by the the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY13A040006), and the K. C. Wong Magna Foundation in Ningbo University.
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
References [1] I. Stiopkin, Tobias Brixner, Mino Yang, G.R. Fleming, J. Phys. Chem. B 110 (2006) 20032. [2] E. Collini, C.Y. Wong, K.E. Wilk, P.M.G. Curmi, P. Brumer, G.D. Scholes, Nature 463 (2010) 644. [3] P. Huo, D.F. Coker, J. Phys. Chem. Lett. 2 (2011) 825. [4] T. Brixner, J. Stenger, H.M. Vaswani, M. Cho, R.E. Blankenship, G.R. Fleming, Nature 434 (2005) 625. [5] M. Cho, T. Brixner, I. Stiopkin, H. Vaswani, G.R. Fleming, J. Chin. Chem. Soc. 53 (2006) 15e24. [6] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, vol. 29, Oxford University Press New York,, 1995. [7] P. Hamm, M. Zanni, Concepts and Methods of 2D Infrared Spectroscopy,
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
5
Cambridge University Press, 2011. S. Mukamel, Annu. Rev. Phys. Chem. 51 (2000) 691. Y.-C. Cheng, R.E. Blankenship, G.R. Fleming, Nature 446 (2007) 12. W.G. Noid, R.F. Loring, Mol. Phys. 103 (2005) 3071. V. Singh, A. Fidler, B. Rolczynski, G. Engel, J. Chem. Phys. 139 (084201) (2013). S. Woutersen, P. Hamm, J. Phys. Chem. B 104 (2000) 11316. S. Park, K. Kwak, M. Fayer, Laser Phys. Lett. 4 (2007) 704. A. Ishizaki, Y. Tanimura, Chem. Phys. 347 (2008) 185. L. Seidner, G. Stock, W. Domcke, J. Chem. Phys. 103 (1995) 3998. L. Chen, R. Zheng, Q. Shi, Y. Yan, J. Chem. Phys. 132 (2010) 227. A.V. Pisliakov, M. Toms, G.R. Fleming, J. Chem. Phys. 124 (2006) 7749. W.M. Zhang, T. Meier, V. Chernyak, S. Mukamel, J. Chem. Phys. 108 (1998) 7763. A. Ishizaki, Y. Tanimura, J. Phys. Soc. Jpn. 74 (2005) 3131. A. Ishizaki, Y. Tanimura, J. Chem. Phys. 125 (2006) 10369. I. Akihito, T. Yoshitaka, J. Phys. Chem. A 111 (2007) 9269. M.M. Sahrapour, N. Makri, J. Chem. Phys. 132 (2010) 134506. X.-T. Liang, J. Chem. Phys. 141 (044116) (2014). X.-T. Liang, Chin. Phys. Lett. 29 (084211) (2012). C. Minhaeng, H.M. Vaswani, B. Tobias, S. Jens, G.R. Fleming, J. Phys. Chem. B 109 (2005) 10542. H.G. Duan, A.G. Dijkstra, P. Nalbach, M. Thorwart, Phys. Rev. E 92 (042708) (2015). Y. Tanimura, S. Mukamel, J. Chem. Phys. 99 (1993) 9496. Y. Tanimura, Y. Maruyama, J. Chem. Phys. 107 (1997) 1779. J. Wu, J. Cao, J. Chem. Phys. 115 (2001) 5381. W.G. Noid, G.S. Ezra, R.F. Loring, J. Chem. Phys. 120 (2004) 1491. W.G. Noid, G.S. Ezra, R.F. Loring, J. Phys. Chem. B 108 (2004) 6536. N. Makri, J. Math. Phys. 36 (1995) 2430. E. Sim, N. Makri, J. Phys. Chem. B 101 (1997) 5446. E. Sim, N. Makri, Comput. Phys. Commun. 99 (1997) 335. X.-T. Liang, Phys. Rev. B 72 (2005) 245328. M. Ali, Phys. Lett. A 378 (2014) 2048. F. Benatti, F. Carollo, R. Floreanini, H. Narnhofer, Phys. Lett. A 380 (2016) 381. J. Piilo, K. Harkonen, S. Maniscalco, K.A. Suominen, Phys. Rev. A 79 (062112) (2009).
Please cite this article in press as: X.-T. Liang, Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates, Computational Condensed Matter (2016), http://dx.doi.org/10.1016/j.cocom.2016.09.006