Non-Markovian Quantum Jump with Generalized Lindblad Master Equation
The Monte Carlo wave function method or the quantum trajectory/jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. In this paper, we extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Markovian jump approach and the computational efficiency of this method are also discussed.
pacs:05.60.Gg, 03.65.Yz, 42.50.Lc
Since the pioneering work of Albert Einstein, who explained the phenomenon of dissipation and Brownian motion in his annus mirabilis of 1905 by use of statistical methods, a rich variety of methods to tackle quantum fluctuations and quantum dissipation in open systems has been proposedquantumnoisy (); dissipation (). Among them the quantum master equation (QME) approach and the quantum Langevin description (QLE)quantumoptics () are two of powerful functional integral techniques for the study of time evolution of open quantum systems. The quantum master equation can be divided into two categories: Markovian and non-Markovian. The Markovian master equationLindblad () (especially in the Lindbald form) can be derived with the weak coupling limit(or the Born approximation) and the Markovian approximation. It can be solved analyticallysolvemasterequ () for some special cases, but for most cases we have to solve and simulate it numerically by the Monte Carlo wave function method or quantum trajectory/jump approachwavefunction (); geometric (); IJQI (); Berryphaseyi (); superdensecoding (); Teleportation (). This method is very effective for qubit systems even with large number of qubits, say Teleportation ().
However, the dynamics of an open system is not always Markovian. Strong system-environment couplings, correlation and entanglement in the initial state and structured reservoirs may lead the dynamics far from Markovian. Many methods have been proposed to describe the non-Markovian process, including the Lindblad equation with time dependent decay ratesBPbookprojection (), generalized Lindblad equationprojection () obtained from the correlated projection superoperator techniquesprojection3 (); projection2008 (), phenomenological memory kernel master equationexpmemoryhazard (); expmemorypositivity () and the post-Markovian master equationpostMarkovian (); postMarkoviansolutionPRA (); postMarkoviansolutionHuang (). The first two methods are local in time while the last two involve an integral of time. For the first method, the only difference from the Markovian master equation is that the decay rates in the equation are time-dependent. These decay rates may take not only positive values but also negative ones. When decay rates are positive, the Markovian Monte Carlo wave function method can directly be used. However, the mehtod is not available when the decay rates are negative. This problem was solved in Ref.nonMQJ () by introducing reversed jumps.
The generalized Lindblad master equation can well describe the dynamics of an open system beyond the Markovian limit, especially it is very effective for an environment composed of spinsspinbath1 (); spinbath2 (); spinbath3 () and structured reservoirsprojectionmodel (). However the extension of the Monte Carlo simulation to this equation remains untouched. In this paper, we will explore the unraveling and quantum trajectory approach for the generalized Lindblad equation. The structure of this paper is organized as follows. In Sec.II we briefly review the generalized Lindblad equation. In Sec.III we give the unraveling of this equation and generalize the Monte Carlo method to this equation. Two examples are presented in Sec.IV. Finally, we conclude our results in Sec.V.
Ii Generalized Lindblad Master Equation
The equation that governs the dynamics of an open quantum system can be derived by means of the projection superoperator techniqueBPbookprojection (); projection3 (). The form (Markovian or non-Markovian) of the master equation crucially depends on the approximation used in the derivation, reflecting in the projection superoperator chosen. When we project the total system state into a tensor product, we can obtain the Markovian master equation, whereas a non-Markovian master equation can be obtained when we use a correlated projection. The following is the master equation derived by this method and it is called the generalized Lindblad master equationprojection (),
where are Hermitian operators and are arbitrary system operators depending on the form of system-environment interactions. If we have only a single component , this equation obviously reduces to the ordinary Markovian master equation. In this paper we will focus on the case where we have at least two components. The state of the reduced system in this case is , we remind that .
Iii Quantum Jump
For clarity, we define the jump operators and non-jump operators , where the non-Hermitian effective Hamiltonian is given by . There are two subscripts and one superscript for the operator . The first subscript denotes the index of component where the system in, while the second subscript denotes the index of component for the operation acting on; the superscript represents the jump mode. Initially we assume that each operator can be written as , where is a non-normalized wave function. After an infinitesimal time , it evolves into the following state
where the new states are defined by
is the weight for the component that satisfies
Note that the jumps for depend on the other components of the reduced density matrix . This makes our method different from the traditional quantum jump method.
Simple algebra shows that in the limit , Eq(8) reveals Eq.(1). The evolution governed by Eq.(1) can be simulated numerically by the so-called Monte Carlo wave function approach according to the unraveling given above. We start the time evolution from the state , where are the components for . At time , where is much smaller than the timescale relevant for the evolution of the density matrix, a random number which is randomly distributed in the unit interval is used to determine the jump. Note that all the components are controlled by this random number. For each component , if , it jumps to , if , it jumps to , and so on. These jumps are all operated on the component ; if , it jumps to the component , namely . Jumps to the other components can be established in a similar way. If , a non-jump takes place and the state ends up in . This operation is acted on the component itself. We define a generalized jump superoperator , which denotes all jumps for all the components controlled by this random number. We repeat this process as many times as for all the components, where is the total evolution time. We call this single evolution a generalized quantum trajectory. This trajectory contains all the components of the density matrix. Given an operator , we can write its mean value as an average over trajectories as
In this section, we use the model and the generalized master equation given in Refs.projectionmodel () and spinbath2 () as two examples to illustrate our method. First consider a two-state system coupled to an environment. The environment consists of a large number of energy levels which are arranged into two energy bands with the same energy spacing(see Fig.1). The lower energy band contains levels while the upper one levels. This model can be understood as a ”many level” environment or ”container”, of which only the relevant parts of the spectrum enter the model. For details of this model, we refer the reader to model2 (); model3 (). The total Hamiltonian for a qubit coupled to such an environment in Schrödinger picture is projectionmodel () with(we set )
where the index denotes the levels of lower energy band and denotes the levels of upper band, and are Pauli operators. is the overall strength of the interaction, are coupling constants, they are independent of each other and are identically distributed, satisfying
According to , one can transform the problem into the interaction picture and, with the help of projection superoperator technique, obtain the non-Markovian evolution equation as
With definitions of and , , the two non-normalized density matrixes can be obtained by , where is the total density matrix for the system and environment. The reduced density matrix for the system is then given by . We note that in Eq.(10), there are no environment operators other than the two (c-number) parameters . The initial state of the environment is taken into account by means of the distribution of initial , its effect on the system dynamics was plotted in Figs.2 and 3. This equation can be written in the form of Eq.(1) by setting , , , and . In this model, there is only one jump operator for each component, i.e. and , and non-jump operators with and .
We consider two types of initial condition in the following simulation. First, only the lower band of the environment is populated, i.e. . Under this condition, the reduced system can be solved analytically. Another case is, the two bands of the environment are all populated. With this initial condition, we solve the master equation numerically. In both cases, we choose an initial states and for the system, where and denote the excited state and ground state, respectively. We compare the analytic solution and the numerical simulation(solve the equation by Runge-Kutta method) to the results obtained from the quantum jump/trajectory approach in Figs.2 and 3. The trajectory number in this quantum jump approach is . We can see from the figures that the quantum trajectory approach correctly reproduces the system evolution. The errors are sufficiently small, although we choose a small number of trajectories, showing that this method is efficient.
Another example is a qubit coupled to a spin bathspinbath2 (). The full system consists of a central spin interacting with a bath of spins. Such a system can be described by
where denotes the Pauli matrix for the central spin that is the system we are interested in, and stands for the -th spin in the bath. After defining an unperturbed part , where , the Hamiltonian can be transformed into the interaction picture. Assuming the parameters are real and time independent, the master equation reads
where , is the density matrix for the total system(the central spin plus the bath), is a projection superoperator that projects the -component of the bath angular momentum into an eigenvector with eigenvalue . We take as an example, then the density matrix of the central spin has three components, denoted by , respectively. Each component has two jump operators which act on the other two components, and a non-jump operator, which acts on itself. The comparison between directly numerical simulations (by Runge-Kutta) and quantum trajectory method is shown in Fig.4. Here the trajectory number is chosen to be . We can find that as the number of jump operators and components increases, the number of quantum trajectory, with that we can obtain a correct result, increases.
V Conclusion and Discussion
In this paper, we have developed an efficient unraveling for the generalized Lindblad master equation. Based on this unraveling, a generalized Monte Carlo wave function method is presented. It is worth addressing that in this Monte Carlo wave function method, we need only to store non-normalized wave function, i.e. length- vectors ( denotes the number of the components for the reduced density matrix and stands for the dimension of the Hilbert space) instead of the density operator, which are matrices, hence this method saves the computer time and space. The difference between the ordinary quantum jump method and the present one is that the latter describes a non-Markovian dynamics. In addition, the point that each component of the density matrix is non-normalized, and jumps along the component depend on the component other than is also different. By successfully simulating the coupling among those components, this method can simulate the non-Markovian dynamics efficiently. Further examination shows that the computational complexity increases with the number of the components. The increased complexity due to the increase of the components and jump operators can by analyzed as follows. Assume the jump operators and the number of jump operators are restricted to be the same for each component, the possible jump mode for , or the number of the generalized jump superoperators is
Here is the number of jump operators for each component(including the non-jump operator). The role that plays is similar to the number of the jump operators in the ordinary Markovian master equation. It is well known that one downside of the quantum jump approach is the complexity growth as the jump operators proliferate. From Eq.(13), we can find that this downside still exists in the presented method. Still, our method is effective when one simulates the decoherence governed by the non-Markovian master equation, as well as for a system with Hilbert space of high dimension.
This work is supported
by NSF of China under grant Nos. 60578014 and 10775023.
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