Quantum Langevin approach for non-Markovian quantum dynamics of the spin-boson model
One long-standing difficult problem in quantum dissipative dynamics is to solve the spin-boson model in a non-Markovian regime where a tractable systematic master equation does not exist. The spin-boson model is particularly important due to its crucial applications in quantum noise control and manipulation as well as its central role in developing quantum theories of open systems. Here we solve this important model by developing a non-Markovian quantum Langevin approach. By projecting the quantum Langevin equation onto the coherent states of the bath, we can derivie a set of non-Markovian quantum Bloch equations containing no explicit noise variables. This special feature offers a tremendous advantage over the existing stochastic Schrödinger equations in numerical simulations. The physical significance and generality of our approach are briefly discussed.
pacs:03.65.Yz, 42.50.Lc, 05.40.Ca
Quantum Langevin equation (QLE) provides a direct depiction of the temporal behaviors of physical observables under the influence of a bath of quantum particles qle1 (); new1 (); new2 (); qle1.5 (); qle3 (); qle4 (). As such, QLE has many important applications in quantum optics new1 (), the input-output theory new3 (), and the quantum dynamics of dissipative atoms qle6 (); qle6.5 (); qle6.75 (); qle7 (). For deriving a generic Langevin equation, however, Markovian approximation was usually employed to arrive at a tractable equation of motion. QLE beyond Markovian approximation can be also formulated to study the intriguing non-Markovian dynamics of damped quantum systems and the Brownian motion systems qle7.5 (); qle8 (); qle9 (); qle10 (); qle11 (); qle12 (). A methodGri1 (); Gri2 (); Gri3 () based on the Mori expansionMori1 (); Mori2 () can solve these kind of problems conveniently. The main idea of this method is to expand the time dependent operator with a set of time independent basis operators. This set of basis and the corresponding coefficients are govern by two recurrence relations.
In the last decade, the so-called non-Markovian quantum state diffusion (QSD) equation Diosi (); Strunz1 (); Yu1 () has been formulated nonperturbatively, so it can apply to the cases with strong couplings between systems and environments (see, e.g., Wiseman (); Strunz2 (); Jing1 ()). The non-Markovian QSD has provided a powerful tool in numerically simulating many interesting physical models N-level (); N-qubit (). In particular, high-order numerical methods for the non-Markovian QSD Strunz3 (); ZCL (); dwl () have been developed very recently, making some previously intractable problems becoming numerically tractable. In fact, the QSD equation is a stochastic Schrödinger equation and it is solved by invoking the noise realizations. For the important case of spin-boson model sb1 (), however, matters are not as simple as the form of this model due to the fact that the spin-boson model does not admit an analytical treatment and an efficient numerical simulation is prohibited without including the higher-order perturbations Strunz3 (); ZCL (); dwl ().
In this paper, we develop a stochastic quantum Langevin approach to solving non-Markovian quantum dynamics of the spin-boson model. This model which is the multi-mode case of the quantum Rabi modelrabi1 (); rabi2 (); rabi3 () involves non-conserving processes due to the counter-rotating terms. Consequently, it poses a long-standing difficult problem in studying non-Markovian quantum dynamics sb1 (). By projecting the non-Markovian QLE onto the coherent states of the bath, we convert the operator QLE into a c-number stochastic QLE, which is formally analogous to the non-Markovian QSD equation. Therefore, the useful techniques developed for the QSD can apply to the c-number stochastic QLE as well. Remarkably, we find that the stochastic QLE can be further reduced to a set of simple non-Markovian quantum Bloch equations without involving any noise variables. This provides a much more efficient method to solve the non-Markovian quantum dynamics of the spin-boson model. As shown below, the method developed here is quite general, so it may offer significant numerical advantages for simulating open quantum systems coupled to bosonic environments when higher-order perturbation is unavoidable.
The paper is organized as follows. In Sec. II, we obtain a stochastic QLE by projecting the non-Markovian QLE onto the coherent states of the bosonic bath. Then, in Sec. III, we convert the stochastic QLE into a c-number stochastic QLE, which is formally analogous to the non-Markovian QSD equation. In Sec. IV, we further reduce the c-number stochastic QLE to a set of simple non-Markovian quantum Bloch equations. The Extensions to the cases of complex correlation function and finite temperature are discussed in Sec. V and VI, respectively. Finally, Sec. VII gives the conclusion of our work.
Ii Stochastic QLE
The spin-boson model is described by , with (setting )
Here is the Hamiltonian of the uncoupled spin and multi-mode bosonic bath, models the interaction between the spin and the bosonic bath, , and are Pauli operators, and () is the th-mode bosonic creation (annihilation) operator of the bath. We assume that the state of the total system is initially factorized as , where the bosonic bath is in the vacuum state (i.e., at zero temperature).
The interaction Hamiltonian can be rewritten as the sum of rotating and counter-rotating terms,
with . The counter-rotating terms and break the conservation of excitation number, giving rise to high-order noise appearing in the stochastic equation of quantum dynamics Diosi (); Strunz1 (); Yu1 ().
Starting from the Heisenberg equations of the Pauli operators and the bosonic operators of the bath, one obtains
are Pauli operators in the Heisenberg picture, and
are the field operators in the Heisenberg picture. The field operators and in Eq. can be formally solved
Substituting the field operators in Eq. with the formal solution in Eq. we can achieve the QLE,
with is the correlation function of the bath, defines a noise operator, and
with . In this QLE, both and act as “random noises” acting on the spin.
Here we first consider the case of real correlation function , so that in Eq. (4). Note that the typical Ornstein-Uhlenbeck correlation function is indeed a real function. We define Bargmann coherent states for the bosonic bath,
When projected onto the Bargmann coherent states, the QLE in Eq. (4) is then converted to
This is a stochastic QLE with the noise . In Eq. (7), , with , and the functional chain rule,
is used. Note that
Iii C-number stochastic QLE
To convert the stochastic equation of operators [i.e., Eq. (7)] into a c-number equation, we introduce the expectation value of an operator as . Then, we have
Define , where denotes the transpose of a matrix. Equation (9) can be written in a matrix form as
Formally, this c-number stochastic QLE is analogous to the non-Markovian QSD equation governing non-Markovian quantum trajectories Diosi (). The difference here is that the QSD equation is a stochastic differential equation for quantum states of the system (i.e., a stochastic Schrödinger equation), while the c-number stochastic QLE in Eq. (10) corresponds to a stochastic differential equation of physical variables.
Here we introduce operator by
Note that although we use the notation which is similar to the operator used in QSD approach, their meanings are different. Here is defined for an arbitrary operator, rather than for a quantum state. Now we can write the c-number stochastic QLE in a time-local form,
Also, using Eq. (13) and the relation
we obtain the equation for operator,
As in Ref. Wiseman, , the initial condition of the operator can be derived as .
Iv Non-Markovian quantum Bloch equation
To obtain the desired quantity
where , one can numerically solve Eq. (13) for each realization of the noise and then implement the statistical average, as in the case of numerically solving QSD equation. However, when higher-order perturbation is involved in the QSD, one must pay the price of long computation time in order to achieve accurate results. Below we show that with our QLE approach, this simulation process can be significantly sped up.
By directly implementing statistical average on Eq. (13), we have
In Eq. (16), can be written as
where . Because , we have
where we have used the relation .
It is known that the operator can be expanded as Yu1 ()
Eq. (16) is finally reduced to our central result
Here we call it a non-Markovian quantum Bloch equation, in which no noise variables are involved. Now note that only the noiseless term of the functional expansion in Eq. (18) is important in solving the non-Markovian quantum dynamics of the system. Because no noise variables are involved, Eq. (20) can be numerically solved very efficiently.
Figure 1 shows the time evolution of for a bath with the Ornstein-Uhlenbeck correlation function. It can be seen from Fig. 1(a) that when increasing , the environmental memory time decreases and exhibits a clear transition from an oscillation to an exponential decay. Physically, this is to some extent connected to the overdamped oscillator, where increasing the friction on the velocity of the oscillator has the effect of turning an oscillation into an exponential decay.
The results are obtained by solving the non-Markovian quantum Bloch equation (20), with determined by (see Appendix A)
with initial condition . These hierarchical equations do not contain any explicit noise variables. In numerical calculations, one can truncate Eq. (21) at a given hierarchical order by choosing . The results in Fig. 1(a) are very similar to those in Ref. ZCL () obtained using the QSD method, showing apparent non-Markovian behaviors at small values of . In ZCL (), the simulations for the curve with took about 36 days to execute on an Intel core-i7 CPU core, but only a few seconds here by solving the non-Markovian quantum Bloch equation (20) via the noiseless hierarchical equations in Eq. (21). This is because of the numerical efficiency of our method without invoking any noise realizations. While at and 3 deviate from those at and 100, at and 100 look nearly identical [see Fig. 1(b)], revealing fast convergence of our results with the hierarchical order . In contrast, the results of obtained using the QSD method have considerable differences between the and 100 orders of the hierarchical equation (see Fig. 2 in ZCL ()), indicating much slower convergence with there.
Many years ago, a proposalwasmade to convert aQLE with correlated fluctuations into a set of coupled equations Gri1 (); Gri2 (); Gri3 (). It was originally developed to study the QLE with an additive noise (e.g., the quantum Brownian motion is such a case) and then extended to the multiplicative-noise case Gri2 (). Here we study quantum dynamics of the spin-boson model. This model involves a multiplicative noise in the QLE and is a more complex, open problem of quantum statistical physics. The central point of our approach is to reduce the QLE to a simple differential equation with no noise variables, i.e., the quantum Bloch equation in Eq. (20). Moreover, we obtain a set of coupled equations, as in Refs. Gri1 (); Gri2 (); Gri3 (), and then use it to efficiently calculate in Eq. (20) without invoking any noise realizations. This is the key reason our approach has a high numerical efficiency.
V Extension to the case of complex correlation function
When the correlation function is complex, i.e., , an extra term is added to Eq. (13):
As the simplest approximation, one can apply a Markovian approximation only to the term in Eq. (22) by taking in as . Then, , and both the same c-number stochastic QLE (13) and the same non-Markovian quantum Bloch equation (20) are thus obtained.
Also, we can replace in by . Then, , and , with . This approximation can give very accurate results at the early stage of quantum evolution. The term in Eq. (22) can be written as
Thus, Eq. (22) is reduced to
which has the same form as Eq. (13), with only replaced by . Also, we can derive the equation for operator and the non-Markovian quantum Bloch equation, which have the same forms as Eqs. (13) and (20), respectively, but with replaced by as well.
Vi Finite-temperature extension
With the thermo-field method Diosi (); finitet (); finitet2 (), we can map the finite-temperature bath onto a larger zero-temperature bath, where a fictitious bath with Hamiltonian is introduced. The corresponding Hamiltonian of the total system then reads
When applying a Bogoliubov transformation finitet2 () to the system,
where , the composite bath of bosonic operators and initially prepared in a thermal state is equivalently converted to a virtual composite bath of bosonic operators and in the vacuum state , with and . Now, the Hamiltonian of the total system is transformed to
Similar to Eq. (4), the Pauli operators obeys the QLE
with the temperature-dependent noise operator
where , and
is the finite-temperature bath correlation function.
Using a similar procedures above, we can derive the c-number stochastic QLE at a finite-temperature as
is the temperature-dependent noise, and
The term in Eq. (30) is
where , with the Bargmann coherent states defined by
We have developed a quantum Langevin approach to solving non-Markovian quantum dynamics of the spin-boson model. Instead of directly attacking the spin-boson model with our non-Markovian QLE, we arrive at a c-number stochastic QLE through projecting the operator QLE onto the coherent states of the bath. Furthermore, we have shown that the stochastic QLE can be reduced to a non-Markovian quantum Bloch equation. With the noiseless quantum Bloch equation, we can efficiently solve the non-Markovian quantum dynamics of the spin-boson model. In addition, we show that our approach is general enough to include the finite-temperature bath. Since the spin-boson model does not admit a non-Markovian master equation, therefore, generally one cannot arrive at a set of useful Bloch equations desirable from our experience in dealing with Markov systems. We show in this paper that QLE paves a new avenue to bypass the stringent difficulty in deriving the non-Markovian master equations. We expect our stochastic quantum Langevin approach can play an important role for many other open quantum systems.
Acknowledgements.This work is supported by the National Natural Science Foundation of China No. 91421102, the National Basic Research Program of China No. 2014CB921401, and the NSAF No. U1330201. T.Y. is supported by the DOD/AF/AFOSR No. FA9550-12-1-0001, and he thanks CSRC for hospitality during his visit.
Appendix A Derivation of the hierarchical equations in Eq. (21)
with , , and for . Here denotes the permutation of all ’s and .
Let us define an operator
with , and consider a noise characterized by the Ornstein-Uhlenbeck correlation function . It can be derived that
This is the first equation in Eq. (21).
For , it can be derived that