The Virtue of Sacrificing Positivity: a Detailed Assessment of Master Equations
The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Here we want to compare some of them focusing on the accuracy of the reduced dynamics. We use two qubits coupled to a Lorentzian environment in a spin-boson like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield Equation with time dependent coefficients yields significantly better results than other methods. Notably, we confirm that the known positivity problems of the Redfield Equation become relevant only in a regime where the underlying approximations are no longer valid, anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak coupling assumption. Further, we discuss how the Coarse-Grained Master Equation with a suitably chosen coarse graining time, and a related Dynamical Map – two methods guaranteeing positivity of the reduced state – can improve on the widely used rotating wave approximation required for the standard Quantum Optical Master Equation. In particular, we show that for a short bath correlation time, the Coarse-Grained Master Equation outperforms the Quantum Optical Master Equation significantly in the regime of stronger coupling, irrespective of the system Hamiltonian.
The non-unitary dynamics of an open quantum system is of great interest for many fields in physics and chemistry, where environmental effects have to be considered. Solving the microscopic model of the whole, system plus environment, with regard to the exact reduced dynamics is in general still a difficult task (see, e.g. Makri (1995); Thorwart and Jung (1997); Beck et al. (2000); Wang and Thoss (2003); Ishizaki and Tanimura (2005); Tanimura (2006); Suess et al. (2014); Hartmann and Strunz (2017)). However, in the weak coupling regime time local master equations can be derived from the microscopic model resulting in approximate solutions for the reduced dynamics Cohen-Tannoudji et al. (1998); Breuer and Petruccione (2007); Weiss (2008). The great advantage of master equations, being easily solved numerically, is to some extent dissolved by the lack of a criterion to estimate the error of the approximations from within the method.
If one is faced with the task to evaluate the reduced dynamics most accurately, the first choice could be the Redfield Equation Redfield (1957); Breuer and Petruccione (2007); Weiss (2008) since it involves the least approximations to obtain a time local evolution equation for the reduced density matrix. In spite of that, since the Redfield Equation is not of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form, positivity of the reduced state is in general not guaranteed Suárez et al. (1992); Pechukas (1994); unphysical negative eigenvalues of the density operator may occur after some time.
However, this shortcoming does not imply that the dynamics obtained from the Redfield Equation is of little use. In particular, sufficiently weak coupling and a fast decaying bath correlation function (BCF) justify the approximations made, which should render the solution of the Redfield Equation valid within a certain error range. From a practical point of view, this motivates the widely successful application of the Redfield Equation and its variants (see for example Refs. Kohen et al. (1997); Kondov et al. (2001); Egorova et al. (2003); Nitzan (2006); Schröder et al. (2007); Timm (2008); Montoya-Castillo et al. (2015); Jeske et al. (2015); Bricker et al. (2018); Damanet et al. (2019) for more recent quantum chemical, condensed matter and quantum optics applications).
From a more conceptual open quantum system point of view, the lack of (complete) positivity implied by the Redfield Equation results in a rejection of the method Benatti and Floreanini (2005); Rivas et al. (2010). However, positivity preservation can be enforced by further approximations.
The most prominent and seasoned additional approximation is the so-called Rotating-Wave or secular Approximation (RWA) Lindblad (1976); Lidar et al. (2001); Breuer and Petruccione (2007); Weiss (2008). The resulting Quantum Optical Master Equation is of the well known GKSL-type and therefore ensures completely positive dynamics. The applicability of the RWA, however, requires a sufficiently weak coupling of the system to the environment such that the dynamics of the reduced state in the interacting picture takes place on a much larger timescale than the timescale set by the transition frequencies of the system Hamiltonian.
If, for example, the RWA is justified for a single qubit () coupled to an environment, it might not be the case for two slightly detuned qubits () coupled simultaneously to the same environment111Throughout this paper we use units where .. The Hamiltonian of this four dimensional system includes transition frequencies of the order of the detuning which introduces a new, presumably larger timescale, requiring an even weaker coupling to the environment for the RWA to be applicable. Therefore, if the aim is to most accurately obtain the reduced dynamics, it seems a priory not clear whether the Redfield Equation or the Quantum Optical Master Equation suffers a more severe problem.
Consequently, a master equation of GKSL-form which can be derived from the microscopic model without RWA seems desirable. A coarse-graining approach leading to the so-called Coarse-Grained Master Equation Schaller and Brandes (2008); Benatti et al. (2009, 2010); Majenz et al. (2013), with a coarse-graining time as a free parameter, may accomplish that task. It requires no direct RWA which suggests that this method could supersede the usual Quantum Optical Master Equation.
Moreover, a completely positive map, here called ExpZ Map, can be constructed from the -dependent generator of the Coarse-Grained Master Equation, which yields the correct dynamics for short times, while recovering the long time dynamics of the usual Quantum Optical Master Equation Majenz et al. (2013); Rivas (2017).
Naturally, the question for the best method arises. We expect each method to be successful for its suiting parameter regime, where the approximations made are justified. However, we ask for the best method in terms of general applicability for various coupling strengths and bath correlation times, and initial states. We therefore base the decision on the accuracy of the reduced dynamics of a generic system (involving a range of transition frequencies), a case which is more general than the simple spin-Boson model Leggett et al. (1987).
In general it is difficult to quantify the error induced by the approximations made without knowing the exact solution Fleming et al. (2010); Rivas et al. (2010); Lee et al. (2012); Wang and Chen (2013). Only for the special limit of zero coupling strength in combination with a rescaled time, exact results can be obtained, that is, the error of the reduced dynamics vanishes Davies (1974); Fleming and Cummings (2011). In addition, it has been pointed out that a perturbative master equation of order in the coupling strength yields an accuracy for the long time dynamics which is of the order Fleming and Cummings (2011). Obviously, the general statement is of no use to determine the scaling of the error for second order master equations, such as the Redfield Equation and the Quantum Optical Master Equation, since it predicts an error no worse than zeroth order. Further, it cannot help to favor one of the methods considered here, since all the methods are of second order in the coupling strength.
Concerning the applicability of the RWA, recent studies Benatti et al. (2010); Ma et al. (2012); Eastham et al. (2016); Dodin et al. (2018) have pointed out that for two qubits (two oscillators) the RWA yields significant deviation from the exact dynamics favoring the Redfield Equation.
With this work we want to contribute to this discussion by calculating an initial state independent error bound of the dynamics as a function of the coupling strength and bath correlation time. For our investigation we consider the exemplary system of two non-interacting qubits coupled to the same Lorentzian environment in a spin-boson like fashion Leggett et al. (1987) where we distinguish the resonant and slightly detuned case with respect to the qubit frequencies. The extension of the spin-boson model to two qubits is not only interesting from a theoretical point of view by challenging the applicability of the RWA, but also relevant for quantum technologies as it serves as a basic building bloc to implement quantum information tasks Imamoğlu et al. (1999); Clarke and Wilhelm (2008); Ladd et al. (2010).
Ranking the various methods based on the error bound confirms that the Redfield Equation performs best. We show that all methods follow a linear scaling of the error with the coupling strength, except the Coarse-Grained Master Equation where the error saturates at some non-zero value. When using the Redfield Equation with time dependent coefficients, we find that positivity problems of the reduced dynamics do only occur in a regime where the weak coupling formalism is anyway not applicable. Further, we elucidate how the Quantum Optical Master Equation differs for the resonant case from the more general detuned case and show that in the detuned case the correlations between the two qubits are strongly effected by the RWA. We find that the ExpZ Map barely resolves the deficiency of the RWA, whereas the Coarse-Grained Master Equation does so in a stronger coupling and short bath correlation time regime. This property of the Coarse-Grained Master Equation can be understood by the fact that the required time scale separation for the coarse-graining time is independent of the system Hamiltonian and, thus, of its intrinsic time scales. Here, denotes the time scale of the environmentally induced system dynamics, that is, of the system dynamics in the interaction picture (typically a damping time scale).
The manuscript is structured as follows. In Sec. II we briefly introduce the notation for the two-spin-boson model and present its solution in Sec. III. The exact solution in terms of a single pseudo-mode is explained, followed by the various approximative master equations. The results in Sec. IV begin with a general ranking of the methods based on an initial state independent error bound. A discussion of the positivity of the reduced dynamics for the Redfield Equation follows, where the advantage of the time dependent coefficients over the Redfield Equation with asymptotic rates is highlighted. Next, the linear scaling of the error is shown briefly and the influence of the coarse-graining parameter is discussed. Finally, the particular effect of the various master equations on the delicate correlation dynamics within the two qubits is shown. We close with a summary and conclusions.
Ii Two-Spin-Boson Model
The Hamiltonian for two qubits coupled to the same environment, which we will refer to as two-spin-boson model, takes the usual form for an open quantum system with a collective Hermitian operator , coupling the two spins to a common bath of harmonic oscillators,
The coupling constants and the oscillator frequencies define the spectral density (SD) . For the continuous environment we choose a single Lorentzian-like SD222 At first glance it seems unphysical to include negative frequencies. However, the physical meaning can be restored when viewing the Lorentzian SD as mathematical vehicle to conveniently model a non-zero temperature BCF with microscopically defined SD . with central frequency , width and overall coupling strength
The corresponding BCF takes the very pleasant form of an exponential
which allows to easily calculate its half-sided Fourier transform
a function occurring in various master equation approaches.
Iii Exact Numerical Solution and Master Equations
iii.1 Exact Solution - Pseudo Mode
In terms of the reduced dynamics, the general open quantum system Hamiltonian Eq. (1) with Lorentzian SD is equivalent to a pseudo-mode model, where the system couples to a single harmonic oscillator with frequency which in turn is coupled with coupling strength to an environment with a flat SD Imamoglu (1994); Garraway (1997). In other words, the Hamiltonian
with leads to the same reduced dynamics for the system part.
As a consequence of the flat SD for the -modes the imaginary part of the corresponding BCF vanishes and the real part becomes delta-like: . In that case the following master equation of GKSL-type for the state of the system plus pseudo mode is known to be exact (see also Ref. Mazzola et al. (2009)).
After truncating the harmonic oscillator at a suitable level, the solution can be calculated numerically. Tracing out the -mode yields the state for the two-qubit system which serves as reference for the following, when comparing the accuracy of the various perturbative master equations.
iii.2 Master Equations
The goal of the following master equations is to provide an evolution equation for the reduced state of the open quantum system as given in Eq. (1) for an arbitrary SD. Besides sketching derivations, we also examine the effect of the RWA, distinguishing the resonant and detuned two-qubit case. Further, we discuss the implications of the approximations used by the coarse-graining scheme.
iii.2.1 Redfield Equation
To derive the evolution equation for the reduced state Breuer and Petruccione (2007); Kryszewski and Czechowska-Kryszk (2008); Whitney (2008) the Nakajima-Zwanzig projection formalism Nakajima (1958); Zwanzig (1960); Grabert (2006) may be used as starting point. In lowest order of the coupling strength the following expression is obtained
Here and denote the reduced state and the interaction Hamiltonian in the interaction picture. Also an initial product state of the form has been assumed. For the microscopic Hamiltonian Eq. (1) we find explicitly with the force operator . Assuming a thermal initial state , the evolution equation Eq. (7) becomes
with the BCF
For a BCF decaying faster than the dynamical time scale of the reduced state in interaction picture, may well be approximated by under the integral. Finally, substituting and transforming back to Schrödinger picture yields
The remaining interaction picture contribution can be made explicit by trivially rewriting the coupling operator in terms of eigenbase projectors of the system Hamiltonian Breuer and Petruccione (2007). Grouping all terms for a fixed defines
and allows for the decomposition where runs over all possible transition frequencies of . Consequently, for an operator in the interacting picture we can write
Finally, we arrive at the Redfield Master Equation with time-dependent coefficients (Redfield Equation (tdc)):
For the model BCF as given in Eq. (3), a single exponential, the time dependent coefficients can be evaluated explicitly,
For a sufficiently fast decaying BCF the asymptotic values may be used instead of the actual time dependent coefficients (see e.g. Yu et al. (2000); Whitney (2008) for the benefit of keeping the time dependent coefficients). This leads to the Redfield Master Equation with asymptotic coefficients (Redfield Equation (asymp.)). Both variants of the Redfield Equation are not of GKSL-form.
iii.2.2 Quantum Optical Master Equation
With the aim to enforce the GKSL-form for the master equation, Eq. (8) is rewritten with and . As before, for a sufficiently fast decaying BCF the integral can be approximated by replacing with . The resulting equation (8) takes the form
If the magnitude of , which represents the coupling strength, is significantly smaller than the smallest non-zero transition frequency (), it can be argued that so-called secular terms (summands with ) average to zero because of the fast oscillating phase. Keeping only the contributions and replacing by the asymptotic values yields, in the Schrödinger picture, the well-known Quantum Optical Master Equation of GKSL-form
Note, since the so-called Lindblad Operators depend on the eigenvalue of , for the two-qubit-system the equation changes discontinuously with the detuning of the two qubits. In the general case (), the only non-zero read
which accounts for four different terms in the master equation, each one proportional to , or its Hermitian conjugate. In contrast, for the resonant case, that is, , there are only two Lindblad operators
Now additional terms in the master equation appear, proportional to, for example, . Even for infinitesimally detuned qubits, these terms are missing due to the secular approximation which particularly influences the dynamics of the correlations of the two qubits (see Sec. IV.5).
iii.2.3 Coarse-Grained Master Equation
For a small detuning of the two qubits the applicability of the secular approximation is more restricted than in the single qubit case due to slow system dynamics appearing on a time scale . To improve on this limitation while keeping the GKSL-property of the master equation, a coarse-graining procedure has been proposed Schaller and Brandes (2008); Benatti et al. (2009); Majenz et al. (2013).
As in (7), the method is based on a second order expansion of the time evolution operator in the interaction picture with respect to which yields
where is the remaining interaction Hamiltonian in the interaction picture.
Evaluating the trace over the environment on the right hand side is again done approximately by assuming that can be replaced by where has to hold333As for Eq. (7), for thermal states in combination with the usual interaction Eq. (1), this is valid.. We get
This expression suggests to generate the time discrete dynamics by sequentially applying such that – provided the product state assumption is consistent at each time step. In this sense is related to the decay of bath correlations – on that time scale correlations between the system and the environment are expected to become unimportant for the reduced dynamics (see also Ref. Benatti et al. (2009); Majenz et al. (2013) for a discussion on the validity of the Coarse-Grained Master Equation).
However, it has been pointed out that the discrete map is not completely positive Schaller and Brandes (2008) – yet it is a valid GKSL generator. Therefore, if the finite difference may well be approximated by the time derivative of the reduced state, Eq. (20) turns into a master equation of GKSL-type Schaller and Brandes (2008)
Note that in the mathematical limit the double time integral in Eq. (20) scales as . Thus, for the Coarse-Grained Master Equation to be meaningful, a time scale separation as for the Redfield Equation and Quantum Optical Master Equation is required where the coarse-graining time has to satisfy . Again, is the timescale on which the reduced state changes in the interaction picture and is the timescale set by the decay of the BCF.
To actually solve the Coarse-Grained Master Equation numerically, we do not use its formulation in obvious GKSL-form Schaller and Brandes (2008); Benatti et al. (2009). It seems more convenient to rewrite Eq. (21) solely in terms of the coupling operator decomposition
and introduce the coefficients
that depend on the coarse-graining parameter . For the Lorentzian SD given in Eq. (2) the coefficients can be explicitly evaluated. For one finds
As expected, when changing back to the Schrödinger picture with respect to the system, the usual Quantum Optical Master Equation is recovered Schaller and Brandes (2008) for
We kept the Coarse-Grained Master Equation in the interaction picture in order to introduce the Lindbladian which can be used to construct yet another completely positive map.
iii.2.4 ExpZ Map
As seen in Eq. (20), for an initial product state, the expression
which leads to the same short time behavior. For long times, on the other hand, approaches , where is the generator of the Quantum Optical Master Equation (16) in the interaction picture. Consequently, the long-time behavior of the ExpZ Map coincides with the dynamics of the Quantum Optical Master Equation. When solving the ExpZ Map as in the later examples, we directly evaluate the matrix exponential numerically for each time step.
Our main result is the comparison of the various master equations, indicating that the Redfield Equation (tdc) with time dependent coefficients results in the most accurate reduced dynamics for the microscopic Hamiltonian of system and environment. We confirm that even though the Redfield Equation (tdc) is not of GKSL-form, positivity issues of the reduced dynamics do not pose a severe problem because they show up only in a parameter regime where the approximations made are not justified. These two statements ultimately allow for the conclusion that whenever the reduced state violates positivity, the validity of any of the weak coupling approaches considered here is doubtful. Consequently, the lack of positivity preservation of the Redfield Equation (tdc) need not be seen as a shortcoming, but should rather be seen as a welcome feature. The failure to represent the true reduced dynamics cannot be detected by the positivity-preserving equations without reference to other methods.
In order to compare the various approaches here, the exact dynamics (pseudo-mode method) is calculated up to a sufficiently large time which depends on the coupling strength and the time scale of the BCF (see Fig. 1). The propagation time is chosen such that the system-plus-pseudo-mode state is close to the asymptotic state for . More precisely, close refers to the condition for the relative difference where the norm denotes the Hilbert-Schmidt norm. Since we later distinguish the resonant and the detuned case with respect to the two qubit frequencies, it should be noted that the propagation time obtained for the detuned case ( and ) is also used for the resonant case. This is justified because the relaxation towards the steady state is slower for the detuned in comparison to the resonant case.
The asymptotic state is obtained by calculating the kernel of the Lindbladian of the truncated psudo-mode master equation. To obtain convergence with respect to the two-qubit state the truncation level of the psudo-mode is incremented by 4 until the change of the asymptotic two-qubit state is below . Therefore, the final truncation level satisfies . The dependence of the truncation on the coupling strength and the BCF timescale is shown in Fig. 1. The final truncation level for the asymptotic system state is also used when propagating the pseudo-mode master equation in order to obtain the system dynamics which serves as exact reference
iv.1 Error of the Master Equations
To provide error bounds independent of the initial state, we write and use the linearity of the propagator : decomposing an arbitrary initial two-qubit state into tensor products of Pauli matrices Gamel (2016) with and using 444, allows to bound the time dependent deviation as follows
The partial deviation
is calculated independently for each of the 16 combinations by propagating the corresponding “initial condition” (which is a valid quantum state for , only).
To see the main features of the deviation, Fig. 2 provides an exemplary plot with three selected partial deviations and the overall sum. Three points should be noted. First, the perfect mixture as initial condition () yields, at the beginning, the smallest deviation, which, however, quickly reaches its asymptotic value. Second, the largest deviation occur after a short propagation time for initial conditions related to the correlations between the two qubits ( with corresponds to a non-zero Bloch-tensor as initial condition). And third, for the slightly detuned case, the deviation of the Redfield Equation (tdc) is by several orders of magnitude smaller as compared to the Quantum Optical Master Equation.
In order to show quantitatively how the error bound behaves while changing the coupling strength and correlation time we choose the maximum value of the time dependent error bound as measure of accuracy. The value provides the maximum deviation that can occur, independent of the initial state and time555This statement requires that the maximum error was reached within the time interval of propagation which is ensured by choosing for each combination of such that the system has almost reached its asymptotic state (see Fig. 1).. This allows us to compare the accuracy of the various approximative methods while changing the environment.
The results are more than telling for the detuned (Fig. 3) as well as the resonant case (Fig. 4), clearly favoring the Redfield Equation (tdc) over all other approaches considered here. Nonetheless, additional information can be drawn from the above plots.
The lines of constant absolute error bound can well be described by simple scaling laws in the relevant parameter regime (see Fig. 3 and Fig. 4). For the Quantum Optical Master Equation and the related ExpZ Map we find an exponent , corresponding to , as expected from a straight forward weak coupling assumption. The Redfield Equation (tdc), on the other hand, shows an exponent , corresponding to lines . The superiority of the Redfield Equation (tdc) becomes evident through the additional factor . The lines of constant error bound for the Coarse-Grained Master Equation again follow a scaling law with exponent . Each line, however, kinks at a critical correlation time which in turn scales with the coarse graining time (see Fig. 8). The kink reflects an intrinsic error of the Coarse-Grained Master Equation imposed by the condition which is independent of the coupling strength .
While the error bound landscape of the Redfield Equation (tdc) for the detuned case does rarely differ from the resonant case, it significantly changes for the Quantum Optical Master Equation and the related ExpZ Map. The explanation is found in the degeneracy of the resonant system Hamiltonian which results in the Lindblad operators and its Hermitian conjugate. Such Lindblad operators result in different features of the reduced dynamics as compared to the detuned case where the Lindblad operators are solely local operators of the form () and its Hermitian conjugate (see Sec. III.2.2). As a consequence of that, the general detuned-case Quantum Optical Master Equation misses some features of the dynamics of the correlations within the qubit system. More details can be found in Sec. IV.5.
The error bound landscape of the ExpZ Map and the Quantum Optical Master Equation are very similar. The small advantage for the ExpZ Map can be understood by noting that the deviation of the Quantum Optical Master Equation reaches its maximum very quickly (see Fig. 2). The ExpZ Map however, yields the correct dynamics for very short times and approaches the dynamics of the Quantum Optical Master Equation for large times. Therefore the deviation of the ExpZ Map looks like the deviation of the Quantum Optical Master Equation but with a suppressed maximum at the beginning.
Concerning the Coarse-Grained Master Equation, the error bound landscapes are shown for the case where the coarse-graining parameter is of the order of the bare single qubit time scale. It is not affected by the resonance condition of the two qubits, just like the Redfield Equation (tdc). In contrast to the other methods, when decreasing the coupling strength only, the error bound saturates to a minimal value, which in turn depends on the correlation time. This hints again to the fact that for the Coarse-Grained Master Equation to be applicable, the correlations between the system and the environment need to become irrelevant on a faster timescale than the coarse-graining time, irrespective of the coupling strength (see also the discussion in Sec. III.2.3). However, for the detuned case, which is the challenging case for the Quantum Optical Master Equation, Fig. 3 shows that there is a regime (small correlation time and fairly large coupling strength) where the error bound of the Coarse-Grained Master Equation is smaller than the error bound of the Quantum Optical Master Equation and the ExpZ Map, i.e. the Coarse-Grained Master Equation with performs better in that regime (for more details see Sec. IV.4).
The discussion so far has ignored the main criticism concerning the Redfield Equation (tdc), the lack of guaranteed positivity. By choosing a physical state as initial condition () we are able to keep track of the positivity of the reduced state. Further, for a particular initial state the relative error can be calculated which allows a comparison of the methods based on actual error instead of the error bound used earlier. Nonetheless, since it turns out that the relative error landscape for each method is very similar to the error bounds shown in Fig. 3 and Fig. 4 (therefore it is not shown here) the initial condition can be seen as a generic initial condition, not featuring any special behavior with respect to the applicability of the various master equations.
In Fig. 5 the parameter region where the maximum relative error is below is shown for the more challenging case of two detuned qubits ( and ) with initial condition . The earlier picture from the initial state independent discussion is recovered: the Redfield Equation (tdc) covers the largest parameter region. The ExpZ Map performs slightly better than the Quantum Optical Master Equation. The Coarse-Grained Master Equation with outreaches the ExpZ Map for sufficiently short correlations times. To add to this picture, keeping track of the positivity for the reduced dynamics obtained from the Redfield Equation (tdc) reveals that positivity problems do only occur in a parameter region where the Redfield Equation (tdc) becomes significantly invalid. In terms of accuracy, the Redfield Equation (tdc) is clearly to be favored compared to other methods considered here. One can even go further by reading the plot in Fig. 5 such that a positivity violation of the reduced dynamics obtained from the Redfield Equation (tdc) allows to keep track of the validity of the underlying approximations made, without having to refer to the exact solution.
Thus, the criticism directed at the Redfield Equation for not being of GKSL-form may be refuted considerably in the light of its accuracy and, in particular, the benefit of using the positivity violation of the Redfield Equation (tdc) as a criterion for its applicability.
iv.2 The Advantage of Time Dependent Coefficients
It should be emphasized that the error of the Redfield Equation (asymp.) with asymptotic coefficients is slightly larger than the error of the Redfield Equation (tdc) with time dependent coefficients (see Fig. 6). However, the Redfield Equation (asymp.) it still outperforms the other methods under consideration. Notably, even in a regime where the relative error is fairly small, transiently non-positive reduced states may occur when using the asymptotic coefficients. Of course, the order of magnitude of the negative eigenvalue does not exceed the order of the error (see Fig. 6).
The difference between the two variants of the Redfield Equation is shown in Fig. 6, where the maximum relative difference and the minimum negative eigenvalue of the dynamics are plotted for a slice through the parameter space with fixed coupling strength. Although only small in magnitude, non-positive eigenvalues of the Redfield Equation (asymp.) dynamics occur already for correlation times where the relative error is still small. When increasing the correlation time the non-positive eigenvalues increases roughly in the same manner as the relative error. In contrast, for the Redfield Equation (tdc) the non-positivity sets in suddenly.
Examining the time dependence of the smallest eigenvalue (see the right panels in Fig. 6) suggests that there are two causes for the positivity violation. First, using asymptotic coefficients as in the Redfield Equation (asymp.), obviously, is not justified for the initial dynamics on the time scale of the correlation time. As a result, non-positive eigenvalues occur during that initial dynamics. Their magnitude decreases with decreasing correlation time which is in line with the observation that for a delta-like correlation function using the asymptotic coefficients becomes exact. However, the non-positive eigenvalues occurring during the initial dynamics disappear after the correlation time has passed (this initial positivity problem is often discussed in terms of an initial slippage Haake and Lewenstein (1983); Suárez et al. (1992); Gaspard and Nagaoka (1999); Yu et al. (2000); Cheng and Silbey (2005)). Using the time dependent coefficients as in the Redfield Equation (tdc) circumvents this problem entirely (see the useful Ref. Whitney (2008) for a thorough investigation of this phenomenon with analytical results for very short correlation times).
The second reason simply originates from the fact that for larger correlation times (or larger coupling strengths) the perturbative approach of the Redfield Equation in general (both time dependent coefficients and asymptotic rates) becomes invalid, resulting in long lasting violation of the positivity (and accuracy) of the reduced dynamics.
iv.3 Linear Scaling of the Error
Concerning the scaling of the error with the coupling strength, it has been mentioned in the introduction that the scaling has to be as good as zeroth order and cannot be, in general, of second order Fleming and Cummings (2011). On the other hand, since it is also known that the Quantum Optical Master Equation becomes exact in the zero coupling limit Davies (1974) the error has to vanish. In Fig. 7 the scaling of the error is shown for three different environmental correlation times . For all of them the plots suggest a linear behavior for the Redfield Equation (tdc), Quantum Optical Master Equation and ExpZ Map. However, in the case of the Coarse-Grained Master Equation the error seems to decrease as well, when decreasing the coupling strength, until it reaches a finite value. Nonetheless, in particular for short correlation times and suitably large coupling strength , the advantage of Coarse-Grained Master Equation over the Quantum Optical Master Equation becomes evident (see also Sec. IV.4). In any case the Redfield Equation yields considerably better results.
iv.4 The Coarse-Graining Time
From a mathematical point of view, the coarse-graining parameter can be chosen freely. However, we have already stressed earlier (Sec. III.2.3, see also Ref. Benatti et al. (2009); Majenz et al. (2013)) that in order to relate the resulting dynamics to the microscopic model, has to fulfill two conditions. By physical means the condition justifies the product state replacement of the total state after the first time step and, thus, allows to iteratively propagate subsequent time steps Majenz et al. (2013). The other condition , where , ensures sufficiently slow system dynamics in the interaction picture, such that the finite difference is well represented by the derivative Benatti et al. (2009); Majenz et al. (2013).
Notably, the time scale set by the energy differences of the system Hamiltonian does not play a role. Consequently, for suitable environments, where the above time scale separation holds, the Coarse-Grained Master Equation is applicable irrespectively of the system Hamiltonian and, thus, provides a master equation beyond the RWA.
The influence of the coarse-graining parameter on the error landscape is shown in Fig. 8. To examine the effect of the “coarse-graining” of the Coarse-Grained Master Equation we also show the error landscape where the exact reduced state averaged over the coarse-graining parameter in the interaction picture
is used as reference. The average ensures that which also serves as initial condition for the Coarse-Grained Master Equation. Fig. 8 shows that there is no significant difference in the overall error behavior between the two cases (upper row: , lower row ). However, minor differences can be noted in the regime where the error is already small. In that case, the dynamics obtained from the Coarse-Grained Master Equation matches the -averaged exact dynamics better than the non-averaged exact dynamics.
Additionally, the maximum relative difference landscape shown in Fig. 8 reveals that for a particular coarse-graining time and a fixed correlation time the Coarse-Grained Master Equation will not become exact in the limit of zero coupling strength. Instead, an asymptotic non-zero error remains which is due to the only approximately met condition . Further, the plots in Fig. 8 show explicitly that for a very small correlation time, such that a rather small coarse-graining time is justified, the Coarse-Grained Master Equation is also applicable for somewhat stronger couplings, a regime in general not accessible by the Quantum Optical Master Equation. This statement will become more explicit in the example dynamics shown in the following.
iv.5 Influence of the Secular Approximation on the Qubit Correlations
Recall, for the detuned case the Lindblad operators read , and their Hermitian conjugate. Viewing as a parameter of the corresponding Quantum Optical Master Equation (fix the form of the Lindblad operators), the resonant case can also be treated with that Quantum Optical Master Equation. On the other hand, for the resonant case the Lindblad operators and its Hermitian conjugate can be derived explicitly, resulting in a different master equation of GKSL-form. The difference of the two variants becomes obvious by realizing that the Lindblad operators enter the master equation quadratically. For example, the Lamb-shift Hamiltonian for the Lindblad operators derived from the detuned case, however used in resonance , reads
In contrast, using the Lindblad operators additional terms occur in the Lamb-shift Hamiltonian
These additional terms, which effectively result in a unitary coupling between the two qubits Benatti et al. (2003); Tana and Ficek (2004), are missing due to the RWA applied in the detuned case. In the same manner, differences between the two variants of the Quantum Optical Master Equation occur also in the dissipator. The non-local structure (in terms of the two qubits) of the additional contribution will particularly influence the dynamics of the two-qubit correlations (see Fig. 9 and Fig. 10).
To summarize, the special Quantum Optical Master Equation derived for the resonance condition includes non-local terms expected to influence the correlation dynamics of the qubits. Once the detuned case is considered, the formalism of the Quantum Optical Master Equation results in an equation without such non-local terms. It is precisely the motivation of the Coarse-Grained Master Equation and the ExpZ Map to overcome this shortcoming Schaller and Brandes (2008); Benatti et al. (2009, 2010); Majenz et al. (2013); Rivas (2017).
In order to show how the various approaches approximate the dynamics, we pick two pairs of and where the differences are sufficiently well visible. In particular we distinguish between the dynamics of the local expectation value and the non-local quantity .
In Fig. 9 the dynamics for a rather weak coupling strength and a correlation time , which is of the order of the single qubit time scale, is shown. For the resonant case, all methods except the Coarse-Grained Master Equation approximate the exact dynamics very well. As expected, for the slightly detuned case, where the detuning results in an additional system time scale slower than the correlation time, the validity of the Quantum Optical Master Equation breaks down. However, the single qubit dynamics is well recovered. Significant deviations are visible for the correlation dynamics of the two qubits. The ExpZ Map smoothly interpolates from the exact dynamics for short times to the values of the Quantum Optical Master Equation for longer times. Concerning the Coarse-Grained Master Equation, the difference to the exact dynamics is equally visible for both, the local and non-local expectation value independently of the detuning. This is plausible, because the coarse-graining time is larger than the correlation time which renders the Coarse-Grained Master Equation to be inaccurate. The Redfield Equation (tdc), however, matches the exact dynamics even for the non-local contribution in the detuned case.
In the next example, the coupling strength is chosen larger while the correlation time becomes shorter . The same statements from the above example hold. In particular the interpolation of the ExpZ Map between the exact dynamics and the values of the Quantum Optical Master Equation is well visible. As a result, both, the Quantum Optical Master Equation and the ExpZ Map, do not account for the slow decay of the correlations. However, as of the shorter correlation time, the Coarse-Grained Master Equation generally produces more accurate results and particularly outperforms the Quantum Optical Master Equation and ExpZ Map on the correlation dynamics in the detuned case. Again, for all examples, the Redfield Equation (tdc) provides the most accurate results.
We have confirmed that in order to best approximate the reduced dynamics of two non-interacting qubits coupled to a common structured (Lorentzian) environment with a time local master equation, the Redfield Equation (tdc) is the method of choice. As indicated by an initial state-independent error bound, the Redfield Equation (tdc) substantially outperforms the other methods considered here (Redfield Equation (asymp.), Quantum Optical Master Equation, Coarse-Grained Master Equation and ExpZ Map). Further, the lack of ensured positivity preservation should not be considered as a bug, but as a feature: it indicates the breakdown of the weak coupling approximation.
In order to contribute to a better understanding of the applicability of the various master equations, we have investigated their error in detail. For the Quantum Optical Master Equation we have explicitly argued – and confirmed by examples – that in the general detuned case of the two qubits, the RWA most significantly effects the correlations between the two qubits.
The two related approaches, the Coarse-Grained Master Equation and the ExpZ Map, do – to some extent – improve on the shortcomings of the Quantum Optical Master Equation as they do not explicitly make use of the RWA while yielding positive dynamics. Our error analysis reveals that the ExpZ Map performs slightly better in terms of the maximum error for the entire dynamics but mimics qualitatively the same error landscape as the Quantum Optical Master Equation. Moreover, we find that whenever the time scale separation is satisfied, the Coarse-Grained Master Equation yields good results, irrespectively of the system Hamiltonian, that is, it does not distinguish between the detuned and resonant case. The coarse-graining parameter -dependent error landscape also qualitatively differs from the Quantum Optical Master Equation correspondent. Exploiting this feature allowed us to explicitly show that there is a region in the parameter space spanned by the coupling strength and correlation time where the Coarse-Grained Master Equation outperforms the Quantum Optical Master Equation significantly.
Although we focus on a particular system of two qubits and a Lorentzian environment, we are confident that our conclusions hold true for generic systems that contain a wide range of transition frequencies.
Fruitful discussions with Kimmo Luoma and Sebastian Diehl are gratefully acknowledged. The computations were performed on a Bull Cluster provided at the Center for Information Services and High Performance Computing (ZIH) at TU Dresden. Support by the IMPRS at the Max Planck Institute for the Physics of Complex Systems (Dresden) and in part by the National Science Foundation under Grant No. NSF PHY-1748958 during time at KITP (UCSB) is gratefully acknowledged.
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