Acessibilidade / Reportar erro

A simple derivation of the Lindblad equation

Uma derivação simples da equação de Lindblad

Abstracts

We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.

Lindblad equation; open quantum systems


Apresentamos uma derivação da equação de Lindblad -uma ferramenta importante no tratamento de evoluções não-unitárias -acessível a estudantes de graduação em física ou matemática com noções básicas de mecânica quântica. Consideramos aqui um caso específico, correspondente a uma situação bem simples, onde o sistema principal interage com um banho de osciladores harmónicos à temperatura nula, com hamiltoniano de interação que se assemelha ao modelo de Jaynes-Cummings. Iniciamos com a equação de Born-Markov e, através do traço parcial dos graus de liberdade do banho, obtemos uma equação na forma de Lindblad. Essa situação específica é bem instrutiva, pois permite verificar que os lindblads representam a contribuição do sistema principal ao hamiltoniano de interação com o banho, e que a aproximação markoviana é vital para o surgimento do lindbladiano. A dedução formal da equação de Lindblad para situações gerais requer o uso do formalismo de semigrupos dinâmicos quânticos e considerações mais abrangentes sobre o ambiente e a temperatura do que as utilizadas aqui.

equação de Lindblad; sistemas quânticos abertos


ARTIGOS GERAIS

A simple derivation of the Lindblad equation

Uma derivação simples da equação de Lindblad

Carlos Alexandre BrasilI,1 1 E-mail: carlosbrasil.physics@gmail.com. ; Felipe Fernandes FanchiniII; Reginaldo de Jesus NapolitanoIII

IInstituto de Física "Gleb Watahgin!!, Universidade Estadual de Campinas, Campinas, SP, Brasil

IIFaculdade de Ciências de Bauru, Universidade Estadual Paulista "lulio de Mesquita Filho", Bauru, SP, Brasil

IIIInstituto de Física de São Carlos, Universidade de São Paulo, São Carlos, SP, Brasil

ABSTRACT

We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.

Keywords: Lindblad equation, open quantum systems.

RESUMO

Apresentamos uma derivação da equação de Lindblad -uma ferramenta importante no tratamento de evoluções não-unitárias -acessível a estudantes de graduação em física ou matemática com noções básicas de mecânica quântica. Consideramos aqui um caso específico, correspondente a uma situação bem simples, onde o sistema principal interage com um banho de osciladores harmónicos à temperatura nula, com hamiltoniano de interação que se assemelha ao modelo de Jaynes-Cummings. Iniciamos com a equação de Born-Markov e, através do traço parcial dos graus de liberdade do banho, obtemos uma equação na forma de Lindblad. Essa situação específica é bem instrutiva, pois permite verificar que os lindblads representam a contribuição do sistema principal ao hamiltoniano de interação com o banho, e que a aproximação markoviana é vital para o surgimento do lindbladiano. A dedução formal da equação de Lindblad para situações gerais requer o uso do formalismo de semigrupos dinâmicos quânticos e considerações mais abrangentes sobre o ambiente e a temperatura do que as utilizadas aqui.

Palavras-chave: equação de Lindblad, sistemas quânticos abertos.

1. Introduction

The Lindblad equalion [1] is lhe mosl general form for a Markovian master equation, and it is very important for the treatment of irreversible and non-unitary processes, from dissipation and decoherence [2] to the quantum measurement process [3,4]. For the latter, in recent applicalions [4,5], lhe Lindblad equalion was used in lhe introduction of time in the interaction between the measured system and the measurement apparatus. Then, the measurement process is no longer treated as instantaneous, but finite, with the duration of that interaction changing the probabilities - diagonal elements of the density operator - associated to the possible final results. On the other hand, in quantum optics, the analysis of spontaneous emission on a two-Ievel system conduels lo lhe Lindblad equalion [6]. AI lasl, in lhe case of quantum Brownian movement, it is possible to lransform lhe Caldeira-Leggell equalion [7] inlo Lindblad wilh lhe addilion of a lerm lhal becomes small in lhe high-lemperalure limil [2]. These are a couple of many applications of the Lindblad equation, justifying its understanding by students in the early levels.

Contrasting against its importance and wide range of applications, its original deduction [1] involves the formalism of quantum dynamical semigroups [8, 9], which is quite unfamiliar to most of the students and researchers. Other more recent ways to derive it involve the use of !tá stochastic calculus [10,11] or, in the specific case of quantum measurements, considerations about the interaction between the system and the meter [12]. Another deduction, where the quantum dynamical semigroups are not explicitly used can be found on Ref. [2]. These methods, their assumptions, their applications, and, more importantly, their physical meanings appear very intimidating to beginning students.

To make the Lind blad equation more understandable, this article presents its deduction in the specific case of two systems: S, the principal system, and B, which can be the environrnent or the measurement apparatus, at zero temperature, with an interaction between them that resembles the one of the JaynesCummings model [2]. Initially we derive the BomMarkov master equation [2] and then we trace out the degrees of freedom of system B. The Lindbladian emerges naturally as a consequence of the Markov approximation. Each Lindblad represents the effect on system S caused by the S -B interaction.

Clearly, the present approach does not prove the general validity of the Lindblad equation. Qur intention is simply to provide an accessible illustration of the validity of the Lindblad equation to non-specialists. The only prerequisite to follow the arguments exposed here is a basic knowledge of quantum mechanics, including a familiarity with the concepts of the density operator and the Liouville-von Neumann equation, at the leveI of Ref. [13], for example.

The paper is structured as follows: in the sec. 2 we derive the Born-Markov master equation by tracing out the degrees of freedom of system B, starting from the Liouville-von Neumann equation; in sec. 3 we derive the Lindblad equation; and in sec. 4 we present the conclusion.

2. The Born-Markov master equation

Let us consider a physical situation where a principal system S, whose dynamics is the object of interest, is coupled with another quantum system B, called bath. Here, HSand HB are, respectively, the Hilbert spaces of principal system S and bath B; the global Hilbert space S + B will be represented by the tensor-product space HSHB. The total Hamiltonian is

where Ĥsdescribes the principal system S, ĤBdescribes the bath B, ĤSB is the Hamiltonian for the system-bath interaction and Band S are the corresponding identities in the Hilbert spaces. Here, we will considerate ĤSand ĤBboth time-independent. For the sake of simplicity, let us ignore the symbol ⊗ and write

Here, α: is a real constant that provides the intensity of interaction between the principal system and the bath. Writing SBfor the global density operator (S + B), the Liouville-von Neumann equation will be:

It is convenient to write Eq. (3) in the interaction picture of Ĥs + ĤB. With the definitions of the new density operator and Hamiltonian:

and

the new equation forp (t) will be

Here and in the following, we will use the time argument explicited ((t)) to indicate the interaction-picture transformation.

We want to find the evolution for S (t) trB {SB(t)} where, according Eq. (5),

Equation (6) is the starting point of our iterative approach. Its time derivative yields

Replacing Eq. (8) into Eq. (6), we have

For the Born approximation, Eq. (9) is enough. Then, we take the partial trace of the bath degrees of freedom,

By the definition in Eq. (4), Ĥ (t) depends on ĤSB , and ĤSBcan always be defined in a manner in which the first term of the right hand side of Eq. (10) is zero. Hence, we obtain

Inlegraling Eq. (11) from t to t' e yields

which shows that lhe difference between S (t) and S (t') is of the second order of magnitude in α: and, therefore, we can write S (t) in the integrand of Eq. (11), oblaining a lime-local equalion for lhe densily operator, without violating the Born approximation

The α conslanl was inlroduced in Eq. (2) only for clarifying the order of magnitude of each term in the iteration and, now, it can be supressed, that is, let us lake α = 1 (full interaction). Thus, let us write

For this approximalion, we can write (t) = S (t) ⊗Binside the integral and obtain the equation that will be used in the next calculations (again, for the sake of simplicily, lei us ignore lhe symbol ⊗)

where we assume that the integration can be extended lo infinily wilhoul changing ils resull. Equalion (14) is lhe BOTn-MaTkov masteT equation [2].

3. Lindblad equation

3.1. The master equation commutator

Let us consider that system-bath interaction is of the following form,

where Ŝ is a general operator that acts only on the principal system S, and is an operator that acts only on the bath B. Now, we consider that Ŝ commutes with ĤS, i.e.,

resulting in

(Ŝ is not affected by the interaction-picture transformation). Let us consider the bath harniltonian defined by a bath of bosons,

where âκ e are the annihilation and creation bath operators, the ωκ are the characteristic frequencies of each mode, and lhe operalor on Eq. (15) defined by

where gκ are complex coefficients representing coupling constantes. Then, in the interaction picture,

Expanding each exponential and using the commutator relalions, Eq. (19) will resull in

The inleraction (15) wilh lhe definilion (18) resembles the Jaynes-Cummings one, who represents a single twoleveI atom interacting with a single mode of the radialion field [2, 141.

Wilh this, lhe commutlator in Eq. (14), , will be evalualed. Firstly,

The gradual expansion of each lerm in Eq. (21) will resull in

and

or, expanding Eqs. (22) and (23) and grouping the similar terms in S and B, we have

and

3.2. The partial trace

Now we are in a position to trace out the bath degrees of freedom in Eqs. (24) and (25). As we can verify with Eq. (20),

With this, then,

and

where, if we use the ciclic properties of the trace,

and

The terms represented by Eqs. (28) and (29) allow us to return to the Eq. (21)

3.3. The expansian af the integrand af the master equatian

With the results of the preceding paragraphs, the integrand in Eq. (14) becomes

For convenience, let us define the functions

Then,

Replacing Eq. (31) in Eq. (14) yields

Actually, the usual Lindblad equation emerges when G(t) = 0 and F(t) = F' (t). ln the following, we make some specifications about the environrnent to discuss these approximations in detail.

3.4. The bath specification

Furthermore, for the initial state of the thermal bath, we consider the vacuum state

The evaluation of the F (t) and G (t) functions defined in Eq. (32) are done considering the (t) and Bdefinitions in Eqs. (20) and (34). By Eq. (20), † (t) is

Then, the partial trace in F (t) and G (t) can be evaluated

and

If we use some bath state basis{lb〉}, Eqs. (36) and (37) become

and

Let us expand † (t) and (t) using Eqs. (20) and (35)

and

Hence, we can rewrite Eq. (40) with the operators on the left of the ák operators. We know that

Then

Therefore, from Eqs. (41) and (43), it follows that

3.5. Transition to the continuum

In the expression of F (t) in Eq. (44), if we adopt the general definition of the density of states as

then the sum over κ can be replaced by an integral over a continuum of frequencies

Let us introduce the new variable Eq. (14), lhal is, we lake

with

yielding

3.6. The Markov approximation

ln the Markov approximation, the limit t → ∞ is taken in the time integral, as we have mentioned regarding Eq. (14), that is, we take

As the integrand oscillates, we will use the device

where P stands for the Cauchy principal parto Then,

3.7. The final form

For a general density of states, F yields

where

As we have verified that G = 0, let us replace Eq. (46) in Eq. (33)

If the density of states is chosen to yield ∈ = 0 (a Lorentzian, for example, where we can extend the lower limit of integration to - ∞), the final result is

Let us, then, return to the original picture. Since

then

Performing lhe sarne operalion on lhe righl-hand side of Eq. (49) gives

Replacing Eqs. (51) and (52) in Eq. (49), we obtain

4. Conclusion

ln summary, in this paper we consider an interaction that resembles the Jaynes-Cummings interaciion [2], Eq. (15), belween a balh and a syslem S, assuming that the operator Ŝ commutes with the system Hamiltonian, ĤS , at zero temperature. We substiluled Eq. (15) in lhe Born-Markov masler Eq. (14) and took the partial trace of the degrees of freedom of B. The T = 0 hypolhesis is necessary lo simplify lhe calculations, making them more accessible to the studenls, simplifying lhe lrealmenl of Eqs. (36) and (37), and avoiding complications such as the Lamb shift in Eq. (46). The Markov approximalion in sec. 3-F was also vilal lo oblain lhe final resull, Eq. (53). Ali lhese simplifications limit the validity of our derivation to more general cases, but it provides a detailed illustration of the physical meaning of each term appearing in lhe Lindblad equalion.

Equation (53) is commonly presented with several Ŝ operators, usually denoted by , in a linear combination of Lindbladian operators. The operators are named Lindblad operators and, in the general case, the Lindblad equalion lakes lhe form

If we consider only the first term on the right hand side of Eq. (54), we obtain the Liouville-von Neumann equation. This term is the Liouvillian and describes the unitary evolution of the density operator. The second term on the right hand side of the Eq. (53) is the Lindbladian and it emerges when we take the partial trace -a non-unitary operation -of the degrees of freedom of system B. The Lindbladian describes the non-unitary evolution of the density operator. By the interaction form adopted here, Eq. (15), the physical meaning of the Lindblad operators can be understood: they represent the system S contribution to the S -B interaction -remembering once more that the Lindblad equation was derived from the Liouville-von Neumann one by tracing the bath degrees of freedom. This conclusion is also achieved with the more general derivation [1,2]. It is important to emphasize that, due to our simplifying assumptions, the summation appearing in Eq. (54) was not obtained in our derivation of Eq. (53).

If the Lindblad operators jare Hermitian (observables), the Lindblad equation can be used to treat the measurement processo A simple application in this sense is the Hamiltonian ĤSz (zis the 2-level z - Pauli mattrix) when we want to measure one specific component of the spin (α, α = x, y, z, without the summation) [3,5]. If the Lindblads are non-Hermitian, the equation can be used to treat dissipation, decoherence or decays. For this, a simple example is the sarne Hamiltonian ĤSz with the Lindblad(_ (), where γ will be the spontaneous emission rate [6].

Acknowledgments

C.A. Brasil acknowledges support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) project number 2011/19848-4, BraziL F.F. Fanchini acknowledges support from FAPESP and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) through the Instituto Nacional de Ciência e Tecnologia -Informação Quântica (INCT-IQ), BraziL R d. J. Napolitano acknowledges support from CNPq, Brazil.

References

[1] Goran Lindblad, Commun. Math. Phys. 48,119 (1976).

[2] Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).

[3] Ian Percival, Quantum State Diffusion (Cambridge University Press, Cambridge, 1995).

[4] Carlos A. Brasil and Reginaldo de J. Napolitano, Eur. Phys. J. Plus 126,91 (2011).

[5] Carlos A. Brasil, Leonardo A. de Castro and Reginaldo de J. Napolitano, Phys. Rev. A 84,022112 (2011).

[6] Michael A. Nielsen and Isaac L. Chuang, Computação Quântica e Informação Quântica (Bookman, Porto Alegre, 2005).

[7] Amir O. Caldeira and Anthony J. Leggett, Physica A 121,587 (1983).

[8] Edward Brian Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).

[9] Robert Alicki and Karl Lendi, Quantum Dynamical Semigroups and Applications (Springer-Verlag, Berlin, 2007).

[10] Stephen L. Adler, Phys. Lett. A 265,58 (2000).

[11] Stephen L. Adler, Phys. Lett. A 267,212 (2000).

[12] Asher Peres, Phys. Rev. A 61,022116 (2000).

[13] Claude Cohen-Tannoudji, Bernard Diu and Franck Lalo, Quantum Mechanics (Wiley, New York, 1977).

[11] Wolfgang P. Schleich, Quantum Optics in Phase Space (Wiley, Berlin, 2001).

Recebido em 2/1/2012

Aceito em 29/8/2012

Publicado em 18/2/2013

  • [1] Goran Lindblad, Commun. Math. Phys. 48,119 (1976).
  • [2] Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  • [3] Ian Percival, Quantum State Diffusion (Cambridge University Press, Cambridge, 1995).
  • [4] Carlos A. Brasil and Reginaldo de J. Napolitano, Eur. Phys. J. Plus 126,91 (2011).
  • [5] Carlos A. Brasil, Leonardo A. de Castro and Reginaldo de J. Napolitano, Phys. Rev. A 84,022112 (2011).
  • [6] Michael A. Nielsen and Isaac L. Chuang, Computação Quântica e Informação Quântica (Bookman, Porto Alegre, 2005).
  • [7] Amir O. Caldeira and Anthony J. Leggett, Physica A 121,587 (1983).
  • [8] Edward Brian Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).
  • [9] Robert Alicki and Karl Lendi, Quantum Dynamical Semigroups and Applications (Springer-Verlag, Berlin, 2007).
  • [10] Stephen L. Adler, Phys. Lett. A 265,58 (2000).
  • [11] Stephen L. Adler, Phys. Lett. A 267,212 (2000).
  • [12] Asher Peres, Phys. Rev. A 61,022116 (2000).
  • [13] Claude Cohen-Tannoudji, Bernard Diu and Franck Lalo, Quantum Mechanics (Wiley, New York, 1977).
  • 1
    E-mail:
  • Publication Dates

    • Publication in this collection
      07 May 2013
    • Date of issue
      Mar 2013

    History

    • Received
      02 Jan 2012
    • Accepted
      29 Aug 2012
    Sociedade Brasileira de Física Caixa Postal 66328, 05389-970 São Paulo SP - Brazil - São Paulo - SP - Brazil
    E-mail: marcio@sbfisica.org.br