- Email: [email protected]
Position dependence of the coupling strength in the Lindblad model of dissipation
Chemical Physics Letters 379 (2003) 113–118 www.elsevier.com/locate/cplett
Position dependence of the coupling strength in the Lindblad model of dissipation Mathias Nest
*
Universit€at Potsdam, Theoretische Chemie, Karl-Liebknecht-Str. 25, 14476 Potsdam, Germany Received 23 July 2003; in final form 6 August 2003 Published online: 4 September 2003
Abstract A formalism is presented which allows to taylor Lindblad operators for open quantum system density matrix theory to produce any desired position dependent coupling strength. Special attention is given to the case, where the coupling strength does not diverge for large distances, as happens in the bilinear coupling model. The method is applied to several common problems and shown to give physically sensible results. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction The Lindblad model [1–4] is, along with Path Integral [5,6] and Redfield techniques [7,8], one of the most widely used approaches to open system quantum dynamics. By far the largest part of the work up to now assumes a coupling between subsystem and bath which is bilinear in the coordinates. This paradigmatic case is useful to derive master equations or study statistical properties of the systems under investigation, but for a large class of real world problems this ansatz is unphysical. E.g., if an atom (system) is scattered from a surface (bath), the coupling should go to zero and not to infinity, for large distances. Similar, if an atom moves parallel to a surface the interaction
*
Fax: +49331977-5058. E-mail address: [email protected] (M. Nest).
should be constant or oscillating, but again not diverging. Additionally, these examples show that the coupling strength between system and bath should be a function of several coordinates. This general multi-dimensional case is beyond the scope of this Letter. However, a few suggestions of how to treat this have been made in [9], and this will be the subject of a future study. In this Letter a formalism is proposed, that can be used to derive dissipative Lindblad operators for any desired position dependence of the coupling strength. It is based on and extends earlier work [9–13], where generalized raising/lowering operators from supersymmetric quantum mechanics [14] (SUSY QM) were used. However, except for a bit of nomenclature, no use of SUSY will be made here. In those previous applications we have shown that these new Lindblad operators give physically very sensible results, like thermalization [10], or sticking probabilities [11,13].
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.08.025
114
M. Nest / Chemical Physics Letters 379 (2003) 113–118
The next section will present the derivation of the formalism, and in Section 3 some solutions for selected problems are discussed. A numerical study of some properties of the newly derived Lindblad operators is given in Section 4. Section 5 concludes this Letter.
2. Formal derivation In general, the master equation for the reduced density matrix of the subsystem has the form [1] i q_ ¼ ½H ; q þ LD ½q: h
ð1Þ
Our discussion will start from the Lindblad form [2–4] of the dissipative Liouvillian LD for energy relaxation due to contact with a Markovian environment at 0 K 1 LD ½q ¼ c aqay ay a; q þ : ð2Þ 2 The generalization to finite temperatures is straightforward [1,10]. The prefactor c sets the characteristic timescale of the dissipation. In order to describe energy dissipation one has to choose raising/lowering operators (RLOs) as Lindblad operators. However, useful RLOs are only available for the Harmonic oscillator. For other systems generalized RLOs (derived in earlier papers [9,10,12]) can be used, which have the following form: 1 i a ¼ pffiffiffiffiffiffi /ðxÞ þ pffiffiffiffiffiffi p ; ð3Þ hx 2m for several systems, like the Morse oscillator. They were obtained using supersymmetric quantum mechanics (SUSY QM) [14]. In this context, the function /ðxÞ is also known as a superpotential. We will continue to use this name, although no use of SUSY QM will be made in this Letter. The expression above reduces to the standard Harmonic RLOs if / / x. As said in the introduction, we will focus here on the position dependence of the dissipation. This is best described by a coupling strength function (CSF), defined by the energy relaxation rate hE_ i of a very narrow gaussian wave packet (Ôprobe functionÕ) centered around x0 :
1 jWi ¼ pffiffiffiffiffiffiffiffiffi pffiffiffiffi exp r p
(
) 2 ðx x0 Þ þ ik0 x ; 2r2
ð4Þ
qðt ¼ 0Þ ¼ jWihWj;
ð5Þ
hE_ iðx0 Þ :¼ TrfH q_ gðx0 Þ ¼ TrfH LD ½qgðx0 Þ:
ð6Þ
A substitution of Eqs. (3) and (2) into Eq. (6) gives [9] rffiffiffiffi _Eðx0 Þ ¼ hc 2 Trf/0 aqay g: ð7Þ m In this derivation it was assumed, that H ¼ hxay a. But as our goal is the derivation of RLOs which can produce an arbitrary position dependence for a given H, this condition can in general not be fulfilled. This is certainly a restriction for the present approach. Nevertheless, as will be shown in the following sections, it works very well even if the RLOs do not factorize the Hamiltonian, as long as only the symmetry and and the general shape of the coupling strength function and the Hamiltonian are similar enough. Eq. (7) gives the relation between the so called superpotential, and the coupling strength function. If one wants to investigate a certain system at hand, one has to solve the inverse problem: the position dependence of the coupling strength between system and bath is given, and one needs to determine /. In order to invert Eq. (7), one has to make the following approximations, based on the narrowness of the chosen gaussian wave packets (Eq. (4)): Z ðxx0 Þ2 1 pffiffiffi /ðxÞe r2 dx /ðx0 Þ; ð8Þ r p 1 pffiffiffi r p
Z
2
/ðxÞ
ðx x0 Þ ðxx20 Þ2 /ðx0 Þ e r dx : 2r2 r4
ð9Þ
Then we obtain rffiffiffiffi x m_ h2 1 2 2 Eðx0 Þ ¼ / ðx0 Þ þ k /0 ðx0 Þ: c 2 2m 0 2r2 ð10Þ This ODE can be integrated straightforwardly by separation of variables.
M. Nest / Chemical Physics Letters 379 (2003) 113–118
x c
rffiffiffiffi m~ 1 3 h2 1 2 EðxÞ ¼ / ðxÞ þ k /ðxÞ: 2 3 2m 0 2r2 ð11Þ
For notational simplicity, the integrated coupling strength function Z x E_ ðx0 Þ dx0 ; ð12Þ E~ðxÞ ¼ has been introduced. Now the values of the superpotential can be calculated by solving the thirdorder polynomial Eq. (11). With the abbreviations rffiffiffiffi h2 1 x m 2 N1 ¼ k ð13Þ N2 ¼ c 2 2m 0 2r2 it can be rewritten as 1 3 / ðxÞ þ N1 /ðxÞ þ N2 E~ðxÞ ¼ 0: 3
ð14Þ
This equation provides the superpotential /ðxÞ for arbitrary CSF E_ ðxÞ. The result can be simplified, if one chooses a Gaussian probe wave function with a minimal width, i.e., r2 ¼ 1=ð2k02 Þ. Then N1 vanishes, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð15Þ /ðxÞ ¼ 3N2 E~ðxÞ: The superpotential is thus independent of the probe function parameters k0 and r, and this is our choice for the applications in the next two sections. In other words, and this is the main result of this Letter, / is basically the cubic root of the integral of the coupling strength function (CSF). The determination of the superpotential concludes the derivation of the generalized RLOs (Eq. (3)), and thus of the dissipative Lindblad functional (Eq. (2)). This can now very flexibly be adapted to various physical situations.
3. Selected solutions In this section we want to calculate the superpotential for a few different physical situations, and discuss the properties of the solutions. In particular, we will look at the harmonic/bilinear case, periodic case, saw tooth, and position independent dissipation. For all numeric calculations
115
we will take the parameters m ¼ 73345 me (argon), c ¼ 1=ð1 psÞ, x ¼ 0:001 Eh =h, and k0 ¼ 10 a1 0 . These are typical values in atomic and molecular physics, and will also be used in the following sections, if not stated otherwise. 3.1. Harmonic case For the Harmonic Oscillator coupled bilinearly to a heat bath according to Eq. (2) one knows that E_ / x2 . If one follows Eq. (15), one has to integrate and take the cubic root of this. Thus one obtains / / x, in accordance with the standard Harmonic RLOs. Inserting E_ ¼ cx2 ;
ð16Þ
via Eq. (12) into Eq. (15) one obtains the superpotential shown in panel a of Fig. 1, a straight line. The parameter c was chosen to be 105 Eh2 =ðha20 Þ. Of course, the formalism presented in this Letter does not aim at the harmonic/bilinear case, which can be treated better with standard methods. Instead, it shall facilitate the treatment of the more complex, nonlinear cases. 3.2. Periodic case The periodic case is especially important in condensed matter physics. A periodically varying strength of dissipation is for example conceivable
0 1 0.8 0.6 0.4 0.2 0 1.4
0.5
1
1.5
2
2.5 0
(a)
5
10
15
20 1.5
(b)
1 0.5 0 1.2
(d)
(c)
1.2 0.8
1 0.8 0.6 0.4 0
0.4 5
10
15
20 0
1
2
3
4
5
6
0 7
Fig. 1. Four superpotentials. (a) Harmonic/bilinear coupling. (b) sin2 periodic coupling stength. (c) Saw-tooth shaped coupling strength. (d) Position independent coupling strength.
116
M. Nest / Chemical Physics Letters 379 (2003) 113–118
for an atom or molecule moving over a crystalline surface. A possible choice is (with c ¼ 105 Eh2 = h, and k ¼ 0:554 a1 0 )
herence due to background radiation or the air pressure in a lab vaccuum. We find (c ¼ 105 Eh2 =h):
E_ ðxÞ ¼ c sin2 ðkxÞ:
ð17Þ
E_ ðxÞ ¼ c
ð18Þ
So basically, the superpotential for position independent dissipation is proportional to x1=3 , see panel c of Fig. 1.
Integration gives c kx sinð2kxÞ E~ðxÞ ¼ ; k 2 4
and the resulting superpotential is shown in panel b of Fig. 1. It is somewhat surprising that the strict periodicity of the CSF is not reproduced in the superpotential. The ÔstepsÕ have the correct distance, but the height of each step gets smaller and smaller. Nevertheless, the use of it in a dissipative Lindblad functional gives physically very reasonable results. The origin of the different heights of the steps is the choice of the lower bound of the integral Eq. (12), which breaks translational symmetry. 3.3. Saw tooth/Quantum ratchet The quantum analogon of a ratchet has seen increased interest during the last few years [15,16]. Here we want to show, that even this somewhat pathological case can be treated with our method. The CSF is (c ¼ 105 Eh2 = h) ! 1 X 1 1 sinðjxÞ E_ ðxÞ ¼ c ; ð19Þ 2 p j¼1 j corresponding to ! 1 X x 1 cosðjxÞ E~ðxÞ ¼ c þ : 2 p j¼1 j2
ð20Þ
The superpotential is shown in panel c of Fig. 1. It shows the same characteristics as the sin2 superpotential.
E~ðxÞ ¼ cx:
ð21Þ
4. Numerical examples In the last section several superpotentials have been discussed, and with them dissipative Lindblad functionals (2) as well. In this section, two properties of these functionals shall be studied numerically. First, we want to see whether the functional does indeed lead to the desired CSF Eq. (17). It might be that the approximations (8) and (9) introduce a large error. Second, we want to look at the predictions of this model for the lifetimes of bound states and compare them with the bilinear/harmonic case. In both cases Eq. (15) is used. 4.1. Coupling strength function To test the CSF, Eq. (7) is evaluated for initial density matrices q ¼ jWihWj (see Eq. (4)) with different centers x0 . The initial momentum of the wave packet was k0 ¼ 30 a1 0 for all center positions. The resulting energy relaxation rate as a function of the position is shown in Fig. 2. The thin line is the desired one, according to Eq. (17). Numerical evaluation (thick line with circles) of Eq. (7) shows that indeed a very close approximation is obtained. The slight deviation is due to the finite width of the gaussian wave packet Eq. (4). 4.2. Lifetimes
3.4. Position independent dissipation It is surprising that this very basic case has never attracted any attention yet. This would be a good choice if one does not have enough information to fit all the parameters of, say, a periodic CSF. Also, it could be a good ansatz for deco-
As an exemplary case we will look at the infinite square well, combined with position independent dissipation. The potential is zero between x ¼ 2, and infinite everywhere else. For the numerical calculations a grid with 256 equidistant points on the interval x ¼ 3 was used. This was also the
M. Nest / Chemical Physics Letters 379 (2003) 113–118
117
/ 1=n2 behavior. It is the analogue to the well known result for a Harmonic oscillator, coupled bilinearly to a harmonic bath, where sn / 1=n, or, more generally, sn ¼ hx=ðcðEn E0 ÞÞ. The shape of the position independent coupling strength and the square well ÔfitÕ to each other. One cannot expect to obtain the result above with arbitrary RLOs.
5. Conclusions
Fig. 2. Desired CSF (thin line), and actual numerically determined CSF (thick line with circles).
interval for the integration of the CSF (Eq. (12)). Again, parameters m ¼ 73345 me , c ¼ 105 , k0 ¼ 30 a1 0 , and c ¼ 1=ð1 psÞ were used. The parameter x was adjusted so that the lifetime of the first excited state (n ¼ 2) comes out as 1=c. The lifetimes were calculated by fitting a function expðt=sn Þ to the time dependent population pn ðtÞ ¼ TrfjnihnjqðtÞg of the nth level, calculated by solving the Liouville–von Neumann equation with the initial condition q ¼ jnihnj. Fig. 3 shows the result. The dots indicate the calculated lifetimes sn , and the straight line shows their perfect
We presented the derivation of Lindblad operators which are taylored to describe an arbitrary position dependent coupling strength. It was shown that the necessary approximations have only a marginal effect. The numerical examples showed physically most sensible properties of the dissipative functionals. Energetically higher lying populations decay with a correspondingly faster rate. However, there are still open questions to be addressed. Both parameters c and c have the meaning of an overall coupling strength. However, they are not redundant, because c appears only in the superpotential part of the operator, and not in front of the momentum operator. However, this is not a new situation, because the prefactor mx in the harmonic RLOs could in principle be chosen independently from the underlying system Hamiltonian (H in Eq. (1)), playing the role of c. If the system is a Harmonic oscillator itself, then there is the natural choice to identify the mass and frequency of the system and of the dissipative operators. In the model presented in this letter however there exists no connection between the coupling strength function and the system Hamiltonian, resulting in the ÔfreeÕ parameter c. We conclude, that the first results reported here are promising, so that the operators derived above may turn out to be a very useful tool for Lindblad type open system quantum dynamics.
Acknowledgements Fig. 3. Lifetimes of bound states of the infinite square well potential, with position independent dissipation.
I would like to thank H.-D. Meyer and P. Saalfrank for valuable discussions and support.
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M. Nest / Chemical Physics Letters 379 (2003) 113–118
References [1] R. Alicki, K. Lendi, in: Lecture Notes in Physics, vol. 286, Springer, Berlin, 1987. [2] G. Lindblad, Commun. Math. Phys. 40 (1975) 147. [3] G. Lindblad, Commun. Math. Phys. 48 (1976) 119. [4] V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17 (1976) 821. [5] R.P. Feynman, F.L. Vernon, Ann. Phys. 24 (1963) 118. [6] U. Weiss, in: Series in Modern Condensed Matter Physics, vol. 2, World Scientific, Singapore, 1993. [7] A.G. Redfield, IBM J. Res. Dev. 1 (1957) 19.
[8] W.T. Pollard, R.A. Friesner, J. Chem. Phys. 100 (1994) 5054. [9] M. Nest, Phys. Rev. A 65 (2002) 052117. [10] M. Nest, P. Saalfrank, Chem. Phys. 268 (2001) 65. [11] M. Nest, P. Saalfrank, J. Chem. Phys. 113 (2000) 8753. [12] M. Nest, P. Saalfrank, J. Chem. Phys. 116 (2002) 7189. [13] M. Nest, H.-D. Meyer, J. Chem. Phys. 119 (2003) 24. [14] F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 269. [15] H. Linke, T.E. Humphrey, A. L€ ofgren, A.O. Sushkov, R. Newbury, R.P. Taylor, P. Omling, Science 286 (1999) 2314. [16] P. H€anggi, P. Reimann, Phys. World 12 (1999) 21.
Position dependence of the coupling strength in the Lindblad model of dissipation Mathias Nest
*
Universit€at Potsdam, Theoretische Chemie, Karl-Liebknecht-Str. 25, 14476 Potsdam, Germany Received 23 July 2003; in final form 6 August 2003 Published online: 4 September 2003
Abstract A formalism is presented which allows to taylor Lindblad operators for open quantum system density matrix theory to produce any desired position dependent coupling strength. Special attention is given to the case, where the coupling strength does not diverge for large distances, as happens in the bilinear coupling model. The method is applied to several common problems and shown to give physically sensible results. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction The Lindblad model [1–4] is, along with Path Integral [5,6] and Redfield techniques [7,8], one of the most widely used approaches to open system quantum dynamics. By far the largest part of the work up to now assumes a coupling between subsystem and bath which is bilinear in the coordinates. This paradigmatic case is useful to derive master equations or study statistical properties of the systems under investigation, but for a large class of real world problems this ansatz is unphysical. E.g., if an atom (system) is scattered from a surface (bath), the coupling should go to zero and not to infinity, for large distances. Similar, if an atom moves parallel to a surface the interaction
*
Fax: +49331977-5058. E-mail address: [email protected] (M. Nest).
should be constant or oscillating, but again not diverging. Additionally, these examples show that the coupling strength between system and bath should be a function of several coordinates. This general multi-dimensional case is beyond the scope of this Letter. However, a few suggestions of how to treat this have been made in [9], and this will be the subject of a future study. In this Letter a formalism is proposed, that can be used to derive dissipative Lindblad operators for any desired position dependence of the coupling strength. It is based on and extends earlier work [9–13], where generalized raising/lowering operators from supersymmetric quantum mechanics [14] (SUSY QM) were used. However, except for a bit of nomenclature, no use of SUSY will be made here. In those previous applications we have shown that these new Lindblad operators give physically very sensible results, like thermalization [10], or sticking probabilities [11,13].
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.08.025
114
M. Nest / Chemical Physics Letters 379 (2003) 113–118
The next section will present the derivation of the formalism, and in Section 3 some solutions for selected problems are discussed. A numerical study of some properties of the newly derived Lindblad operators is given in Section 4. Section 5 concludes this Letter.
2. Formal derivation In general, the master equation for the reduced density matrix of the subsystem has the form [1] i q_ ¼ ½H ; q þ LD ½q: h
ð1Þ
Our discussion will start from the Lindblad form [2–4] of the dissipative Liouvillian LD for energy relaxation due to contact with a Markovian environment at 0 K 1 LD ½q ¼ c aqay ay a; q þ : ð2Þ 2 The generalization to finite temperatures is straightforward [1,10]. The prefactor c sets the characteristic timescale of the dissipation. In order to describe energy dissipation one has to choose raising/lowering operators (RLOs) as Lindblad operators. However, useful RLOs are only available for the Harmonic oscillator. For other systems generalized RLOs (derived in earlier papers [9,10,12]) can be used, which have the following form: 1 i a ¼ pffiffiffiffiffiffi /ðxÞ þ pffiffiffiffiffiffi p ; ð3Þ hx 2m for several systems, like the Morse oscillator. They were obtained using supersymmetric quantum mechanics (SUSY QM) [14]. In this context, the function /ðxÞ is also known as a superpotential. We will continue to use this name, although no use of SUSY QM will be made in this Letter. The expression above reduces to the standard Harmonic RLOs if / / x. As said in the introduction, we will focus here on the position dependence of the dissipation. This is best described by a coupling strength function (CSF), defined by the energy relaxation rate hE_ i of a very narrow gaussian wave packet (Ôprobe functionÕ) centered around x0 :
1 jWi ¼ pffiffiffiffiffiffiffiffiffi pffiffiffiffi exp r p
(
) 2 ðx x0 Þ þ ik0 x ; 2r2
ð4Þ
qðt ¼ 0Þ ¼ jWihWj;
ð5Þ
hE_ iðx0 Þ :¼ TrfH q_ gðx0 Þ ¼ TrfH LD ½qgðx0 Þ:
ð6Þ
A substitution of Eqs. (3) and (2) into Eq. (6) gives [9] rffiffiffiffi _Eðx0 Þ ¼ hc 2 Trf/0 aqay g: ð7Þ m In this derivation it was assumed, that H ¼ hxay a. But as our goal is the derivation of RLOs which can produce an arbitrary position dependence for a given H, this condition can in general not be fulfilled. This is certainly a restriction for the present approach. Nevertheless, as will be shown in the following sections, it works very well even if the RLOs do not factorize the Hamiltonian, as long as only the symmetry and and the general shape of the coupling strength function and the Hamiltonian are similar enough. Eq. (7) gives the relation between the so called superpotential, and the coupling strength function. If one wants to investigate a certain system at hand, one has to solve the inverse problem: the position dependence of the coupling strength between system and bath is given, and one needs to determine /. In order to invert Eq. (7), one has to make the following approximations, based on the narrowness of the chosen gaussian wave packets (Eq. (4)): Z ðxx0 Þ2 1 pffiffiffi /ðxÞe r2 dx /ðx0 Þ; ð8Þ r p 1 pffiffiffi r p
Z
2
/ðxÞ
ðx x0 Þ ðxx20 Þ2 /ðx0 Þ e r dx : 2r2 r4
ð9Þ
Then we obtain rffiffiffiffi x m_ h2 1 2 2 Eðx0 Þ ¼ / ðx0 Þ þ k /0 ðx0 Þ: c 2 2m 0 2r2 ð10Þ This ODE can be integrated straightforwardly by separation of variables.
M. Nest / Chemical Physics Letters 379 (2003) 113–118
x c
rffiffiffiffi m~ 1 3 h2 1 2 EðxÞ ¼ / ðxÞ þ k /ðxÞ: 2 3 2m 0 2r2 ð11Þ
For notational simplicity, the integrated coupling strength function Z x E_ ðx0 Þ dx0 ; ð12Þ E~ðxÞ ¼ has been introduced. Now the values of the superpotential can be calculated by solving the thirdorder polynomial Eq. (11). With the abbreviations rffiffiffiffi h2 1 x m 2 N1 ¼ k ð13Þ N2 ¼ c 2 2m 0 2r2 it can be rewritten as 1 3 / ðxÞ þ N1 /ðxÞ þ N2 E~ðxÞ ¼ 0: 3
ð14Þ
This equation provides the superpotential /ðxÞ for arbitrary CSF E_ ðxÞ. The result can be simplified, if one chooses a Gaussian probe wave function with a minimal width, i.e., r2 ¼ 1=ð2k02 Þ. Then N1 vanishes, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð15Þ /ðxÞ ¼ 3N2 E~ðxÞ: The superpotential is thus independent of the probe function parameters k0 and r, and this is our choice for the applications in the next two sections. In other words, and this is the main result of this Letter, / is basically the cubic root of the integral of the coupling strength function (CSF). The determination of the superpotential concludes the derivation of the generalized RLOs (Eq. (3)), and thus of the dissipative Lindblad functional (Eq. (2)). This can now very flexibly be adapted to various physical situations.
3. Selected solutions In this section we want to calculate the superpotential for a few different physical situations, and discuss the properties of the solutions. In particular, we will look at the harmonic/bilinear case, periodic case, saw tooth, and position independent dissipation. For all numeric calculations
115
we will take the parameters m ¼ 73345 me (argon), c ¼ 1=ð1 psÞ, x ¼ 0:001 Eh =h, and k0 ¼ 10 a1 0 . These are typical values in atomic and molecular physics, and will also be used in the following sections, if not stated otherwise. 3.1. Harmonic case For the Harmonic Oscillator coupled bilinearly to a heat bath according to Eq. (2) one knows that E_ / x2 . If one follows Eq. (15), one has to integrate and take the cubic root of this. Thus one obtains / / x, in accordance with the standard Harmonic RLOs. Inserting E_ ¼ cx2 ;
ð16Þ
via Eq. (12) into Eq. (15) one obtains the superpotential shown in panel a of Fig. 1, a straight line. The parameter c was chosen to be 105 Eh2 =ðha20 Þ. Of course, the formalism presented in this Letter does not aim at the harmonic/bilinear case, which can be treated better with standard methods. Instead, it shall facilitate the treatment of the more complex, nonlinear cases. 3.2. Periodic case The periodic case is especially important in condensed matter physics. A periodically varying strength of dissipation is for example conceivable
0 1 0.8 0.6 0.4 0.2 0 1.4
0.5
1
1.5
2
2.5 0
(a)
5
10
15
20 1.5
(b)
1 0.5 0 1.2
(d)
(c)
1.2 0.8
1 0.8 0.6 0.4 0
0.4 5
10
15
20 0
1
2
3
4
5
6
0 7
Fig. 1. Four superpotentials. (a) Harmonic/bilinear coupling. (b) sin2 periodic coupling stength. (c) Saw-tooth shaped coupling strength. (d) Position independent coupling strength.
116
M. Nest / Chemical Physics Letters 379 (2003) 113–118
for an atom or molecule moving over a crystalline surface. A possible choice is (with c ¼ 105 Eh2 = h, and k ¼ 0:554 a1 0 )
herence due to background radiation or the air pressure in a lab vaccuum. We find (c ¼ 105 Eh2 =h):
E_ ðxÞ ¼ c sin2 ðkxÞ:
ð17Þ
E_ ðxÞ ¼ c
ð18Þ
So basically, the superpotential for position independent dissipation is proportional to x1=3 , see panel c of Fig. 1.
Integration gives c kx sinð2kxÞ E~ðxÞ ¼ ; k 2 4
and the resulting superpotential is shown in panel b of Fig. 1. It is somewhat surprising that the strict periodicity of the CSF is not reproduced in the superpotential. The ÔstepsÕ have the correct distance, but the height of each step gets smaller and smaller. Nevertheless, the use of it in a dissipative Lindblad functional gives physically very reasonable results. The origin of the different heights of the steps is the choice of the lower bound of the integral Eq. (12), which breaks translational symmetry. 3.3. Saw tooth/Quantum ratchet The quantum analogon of a ratchet has seen increased interest during the last few years [15,16]. Here we want to show, that even this somewhat pathological case can be treated with our method. The CSF is (c ¼ 105 Eh2 = h) ! 1 X 1 1 sinðjxÞ E_ ðxÞ ¼ c ; ð19Þ 2 p j¼1 j corresponding to ! 1 X x 1 cosðjxÞ E~ðxÞ ¼ c þ : 2 p j¼1 j2
ð20Þ
The superpotential is shown in panel c of Fig. 1. It shows the same characteristics as the sin2 superpotential.
E~ðxÞ ¼ cx:
ð21Þ
4. Numerical examples In the last section several superpotentials have been discussed, and with them dissipative Lindblad functionals (2) as well. In this section, two properties of these functionals shall be studied numerically. First, we want to see whether the functional does indeed lead to the desired CSF Eq. (17). It might be that the approximations (8) and (9) introduce a large error. Second, we want to look at the predictions of this model for the lifetimes of bound states and compare them with the bilinear/harmonic case. In both cases Eq. (15) is used. 4.1. Coupling strength function To test the CSF, Eq. (7) is evaluated for initial density matrices q ¼ jWihWj (see Eq. (4)) with different centers x0 . The initial momentum of the wave packet was k0 ¼ 30 a1 0 for all center positions. The resulting energy relaxation rate as a function of the position is shown in Fig. 2. The thin line is the desired one, according to Eq. (17). Numerical evaluation (thick line with circles) of Eq. (7) shows that indeed a very close approximation is obtained. The slight deviation is due to the finite width of the gaussian wave packet Eq. (4). 4.2. Lifetimes
3.4. Position independent dissipation It is surprising that this very basic case has never attracted any attention yet. This would be a good choice if one does not have enough information to fit all the parameters of, say, a periodic CSF. Also, it could be a good ansatz for deco-
As an exemplary case we will look at the infinite square well, combined with position independent dissipation. The potential is zero between x ¼ 2, and infinite everywhere else. For the numerical calculations a grid with 256 equidistant points on the interval x ¼ 3 was used. This was also the
M. Nest / Chemical Physics Letters 379 (2003) 113–118
117
/ 1=n2 behavior. It is the analogue to the well known result for a Harmonic oscillator, coupled bilinearly to a harmonic bath, where sn / 1=n, or, more generally, sn ¼ hx=ðcðEn E0 ÞÞ. The shape of the position independent coupling strength and the square well ÔfitÕ to each other. One cannot expect to obtain the result above with arbitrary RLOs.
5. Conclusions
Fig. 2. Desired CSF (thin line), and actual numerically determined CSF (thick line with circles).
interval for the integration of the CSF (Eq. (12)). Again, parameters m ¼ 73345 me , c ¼ 105 , k0 ¼ 30 a1 0 , and c ¼ 1=ð1 psÞ were used. The parameter x was adjusted so that the lifetime of the first excited state (n ¼ 2) comes out as 1=c. The lifetimes were calculated by fitting a function expðt=sn Þ to the time dependent population pn ðtÞ ¼ TrfjnihnjqðtÞg of the nth level, calculated by solving the Liouville–von Neumann equation with the initial condition q ¼ jnihnj. Fig. 3 shows the result. The dots indicate the calculated lifetimes sn , and the straight line shows their perfect
We presented the derivation of Lindblad operators which are taylored to describe an arbitrary position dependent coupling strength. It was shown that the necessary approximations have only a marginal effect. The numerical examples showed physically most sensible properties of the dissipative functionals. Energetically higher lying populations decay with a correspondingly faster rate. However, there are still open questions to be addressed. Both parameters c and c have the meaning of an overall coupling strength. However, they are not redundant, because c appears only in the superpotential part of the operator, and not in front of the momentum operator. However, this is not a new situation, because the prefactor mx in the harmonic RLOs could in principle be chosen independently from the underlying system Hamiltonian (H in Eq. (1)), playing the role of c. If the system is a Harmonic oscillator itself, then there is the natural choice to identify the mass and frequency of the system and of the dissipative operators. In the model presented in this letter however there exists no connection between the coupling strength function and the system Hamiltonian, resulting in the ÔfreeÕ parameter c. We conclude, that the first results reported here are promising, so that the operators derived above may turn out to be a very useful tool for Lindblad type open system quantum dynamics.
Acknowledgements Fig. 3. Lifetimes of bound states of the infinite square well potential, with position independent dissipation.
I would like to thank H.-D. Meyer and P. Saalfrank for valuable discussions and support.
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