2588:, begins with a more general form of an open quantum system and converts it into Lindblad form by making the Markovian assumption and expanding in small time. A more physically motivated standard treatment covers three common types of derivations of the Lindbladian starting from a Hamiltonian acting on both the system and environment: the weak coupling limit (described in detail below), the low density approximation, and the singular coupling limit. Each of these relies on specific physical assumptions regarding, e.g., correlation functions of the environment. For example, in the weak coupling limit derivation, one typically assumes that (a) correlations of the system with the environment develop slowly, (b) excitations of the environment caused by system decay quickly, and (c) terms which are fast-oscillating when compared to the system timescale of interest can be neglected. These three approximations are called Born, Markov, and rotating wave, respectively.
118:
However, any real physical system is not absolutely isolated, and will interact with its environment. This interaction with degrees of freedom external to the system results in dissipation of energy into the surroundings, causing decay and randomization of phase. More so, understanding the interaction of a quantum system with its environment is necessary for understanding many commonly observed phenomena like the spontaneous emission of light from excited atoms, or the performance of many quantum technological devices, like the laser.
1251:
6352:
2714:. This equation, containing an infinite number of degrees of freedom, is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix,
1507:
328:
936:
4477:
The equation is now explicit in the system degrees of freedom, but is very difficult to solve. A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics,
4190:
This equation is exact for the time dynamics of the system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say
2548:
133:, it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has the property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems.
2591:
The weak-coupling limit derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a
Hamiltonian written in terms of operators acting only on the respective subspace of the
117:
In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that there is no decay and phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered.
1086:
6171:
92:
or, actually, the von
Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. The Schrödinger equation deals with
1817:
2076:
6141:
2592:
total
Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is
2308:
1335:
5906:
5224:
6589:
156:
5998:
4731:
4472:
4185:
757:
4947:
2378:
459:, describing the dissipative part of the dynamics. The shape of the jump operators describes how the environment acts on the system, and must either be determined from microscopic models of the system-environment dynamics, or
1690:
3292:
3182:
5815:
3533:
1246:{\displaystyle u^{\dagger }hu={\begin{bmatrix}\gamma _{1}&0&\cdots &0\\0&\gamma _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\gamma _{N^{2}-1}\end{bmatrix}}}
6347:{\displaystyle {\begin{aligned}F_{1}&=a,&\gamma _{1}&={\tfrac {\gamma }{2}}\left({\overline {n}}+1\right),\\F_{2}&=a^{\dagger },&\gamma _{2}&={\tfrac {\gamma }{2}}{\overline {n}}.\end{aligned}}}
3984:
1934:
3814:
4261:
1701:
4800:
659:
6176:
2811:
6034:
4525:
2751:
2157:
3365:
3037:
1954:
1327:
709:
600:
2903:
2163:
3897:
2656:
2712:
496:
5823:
3565:
6382:
733:
533:
453:
381:
6428:
6420:
4857:
4830:
2945:
1847:
1554:
1502:{\displaystyle {\dot {\rho }}=-{i \over \hbar }+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}
972:
323:{\displaystyle {\dot {\rho }}=-{i \over \hbar }+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}
4955:
1584:
1060:
81:
to open quantum systems; that is, systems in contacts with their surroundings. The resulting dynamics is no longer unitary, but still satisfies the property of being
3924:
2569:
The
Lindblad master equation describes the evolution of various types of open quantum systems, e.g. a system weakly coupled to a Markovian reservoir. Note that the
1007:
6594:
Additional
Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based
3843:
931:{\displaystyle {\dot {\rho }}=-{i \over \hbar }+\sum _{n,m}h_{nm}\left(A_{n}\rho A_{m}^{\dagger }-{\frac {1}{2}}\left\{A_{m}^{\dagger }A_{n},\rho \right\}\right)}
6026:
4191:
the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as
2831:
1027:
407:
4862:
6153:
To model both absorption and emission, one would need a jump operator for each. This leads to the most common
Lindblad equation describing the damping of a
2543:{\displaystyle {\dot {X}}={\frac {i}{\hbar }}+\sum _{i}\gamma _{i}\left(L_{i}^{\dagger }XL_{i}-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},X\right\}\right).}
4533:
4269:
3992:
5911:
2577:
necessarily equal to the bare system
Hamiltonian, but may also incorporate effective unitary dynamics arising from the system-environment interaction.
121:
Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment. One of these is the use of the
1596:
3187:
3045:
43:
2352:
of the diagonal form of the
Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.
5235:
3373:
6829:
3932:
1855:
616:
3845:, and at that time there are no correlations between the system and the bath. This implies that the initial condition is factorable as
7345:
4194:
3573:
2557:. Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation is
1812:{\displaystyle {\mathcal {L}}(\rho )=\mathrm {lim} _{\Delta t\to 0}{\frac {\phi _{\Delta t}(\rho )-\phi _{0}(\rho )}{\Delta t}}}
6627:
7300:
7279:
7260:
7237:
7218:
7191:
7142:
6872:
6663:
982:
matrix. The latter is a strict requirement to ensure the dynamics is trace-preserving and completely positive. The number of
4742:
6136:{\displaystyle {\mathcal {L}}(\rho )={F\rho F^{\dagger }}-{\frac {1}{2}}\left(F^{\dagger }F\rho +\rho F^{\dagger }F\right)}
662:
603:
82:
2947:
is the total unitary operator of the entire system. It is straightforward to confirm that the
Liouville equation becomes
2756:
2753:. The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation
2717:
125:, and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the
3297:
7067:
Gorini, V.; Kossakowski, A.; Sudarshan, E.C.G. (1976). "Completely positive dynamical semigroups of N-level systems".
6812:
2071:{\displaystyle {\sqrt {\gamma _{i}}}L_{i}\to {\sqrt {\gamma _{i}'}}L_{i}'=\sum _{j}v_{ij}{\sqrt {\gamma _{j}}}L_{j},}
17:
4481:
2087:
2953:
2303:{\displaystyle H\to H'=H+{\frac {1}{2i}}\sum _{j}\gamma _{j}\left(a_{j}^{*}L_{j}-a_{j}L_{j}^{\dagger }\right)+bI,}
7096:
Banks, T.; Susskind, L.; Peskin, M.E. (1984). "Difficulties for the evolution of pure states into mixed states".
1269:
667:
538:
6158:
2836:
2338:
are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the
3848:
2598:
55:
6392:
6154:
5901:{\displaystyle \langle \Gamma _{i}\Gamma _{j}\rangle =\operatorname {tr} \{\Gamma _{i}\Gamma _{j}R_{0}\}}
5908:
are with respect to the bath degrees of freedom. By assuming rapid decay of these correlations (ideally
7360:
2664:
2553:
A similar equation describes the time evolution of the expectation values of observables, given by the
1065:
The general form is not in fact more general, and can be reduced to the special form. Since the matrix
468:
3541:
6360:
1009:
operators is arbitrary, and they do not have to satisfy any special properties. But if the system is
6584:{\displaystyle {\dot {\rho }}=-i+\gamma _{1}{\mathcal {D}}(\rho )+\gamma _{2}{\mathcal {D}}(\rho ).}
714:
6838:
1517:
979:
505:
418:
336:
7333:
6804:
6607:
6398:
4835:
4808:
3567:
can be substituted back into the
Liouville equation to obtain an exact differo-integral equation
2324:
is a real number. However, the first transformation destroys the orthonormality of the operators
74:
5219:{\displaystyle {\dot {\tilde {\rho }}}=-\sum _{i,j}\int _{0}^{t}dt'\operatorname {tr} _{R}\{]\}}
2908:
89:
78:
6905:
Albert, Victor V. (2018). "Lindbladians with multiple steady states: theory and applications".
1825:
1532:
1074:
944:
460:
51:
2361:
126:
6792:
1563:
1527:
1070:
1032:
106:
7171:
7105:
7076:
7039:
7014:
7002:
6981:
6944:
6757:
6701:
3902:
985:
4478:
wherein correlations within the bath are lost extremely quickly, and amounts to replacing
1029:-dimensional, it can be shown that the master equation can be fully described by a set of
8:
6622:
6617:
3822:
47:
7175:
7109:
7080:
7043:
7018:
6985:
6948:
6925:
ChruĆciĆski, Dariusz; Pascazio, Saverio (2017). "A Brief History of the GKLS Equation".
6761:
6705:
2661:
The dynamics of the entire system can be described by the Liouville equation of motion,
1522:
The maps generated by a Lindbladian for various times are collectively referred to as a
7365:
7197:
7161:
7055:
6972:
Kossakowski, A. (1972). "On quantum statistical mechanics of non-Hamiltonian systems".
6960:
6934:
6906:
6773:
6747:
6717:
6691:
6011:
4942:{\displaystyle {\tilde {H}}_{BS}=\sum _{i}{\tilde {\alpha }}_{i}{\tilde {\Gamma }}_{i}}
2816:
2365:
1012:
392:
130:
98:
7296:
7275:
7256:
7255:. Springer Series in Synergetics (3rd ed.). Berlin Heidelberg: Springer-Verlag.
7233:
7214:
7201:
7187:
7138:
7121:
7117:
6993:
6887:
6868:
6808:
6777:
6659:
6612:
6162:
4726:{\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{]\}}
4467:{\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{]\}}
4180:{\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{]\}}
2554:
31:
7059:
6721:
7179:
7113:
7084:
7047:
6989:
6952:
6860:
6800:
6765:
6709:
5993:{\displaystyle \langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle \propto \delta (t-t')}
2368:
using the following (diagonalized) equation of motion for each quantum observable
7027:
6964:
6679:
6738:
Manzano, Daniel (2020). "A short introduction to the Lindblad master equation".
1685:{\displaystyle \phi _{s}(\phi _{t}(\rho ))=\phi _{t+s}(\rho ),\qquad t,s\geq 0.}
6713:
6595:
6147:
3287:{\displaystyle {\tilde {\chi }}=U_{BS}(t,t_{0})\chi U_{BS}^{\dagger }(t,t_{0})}
3177:{\displaystyle {\tilde {H}}_{BS}=e^{i(H_{S}+H_{B})t}H_{BS}e^{-i(H_{S}+H_{B})t}}
1557:
384:
122:
102:
71:
6956:
6384:
is the mean number of excitations in the reservoir damping the oscillator and
7354:
7183:
2585:
610:
94:
7248:
3819:
We proceed with the derivation by assuming the interaction is initiated at
6864:
3184:
is explicitly time dependent. Also, according to the interaction picture,
602:
describing unitary dynamics, which is the quantum analog of the classical
7327:
7166:
5810:{\displaystyle {\dot {\tilde {\rho }}}=-\sum _{i,j}\int _{0}^{t}dt'\left}
3528:{\displaystyle {\tilde {\chi }}(t)={\tilde {\chi }}(0)-i\int _{0}^{t}dt'}
7001:
Belavin, A.A.; Zel'dovich, B. Ya.; Perelomov, A.M.; Popov, V.S. (1969).
3979:{\displaystyle \operatorname {tr} _{R}{\tilde {\chi }}={\tilde {\rho }}}
7051:
2558:
1929:{\displaystyle \phi _{t+s}(\rho )=e^{{\mathcal {L}}s}\phi _{t}(\rho ).}
7125:
6769:
7272:
Information Dynamics and Open Systems: Classical and Quantum Approach
7152:
Alicki, Robert (2002). "Invitation to quantum dynamical semigroups".
7088:
1587:
27:
Markovian quantum master equation for density matrices (mixed states)
6150:
to model either absorption or emission of photons from a reservoir.
1944:
The Lindblad equation is invariant under any unitary transformation
1062:
operators, provided they form a basis for the space of operators.
150:
can be written as (for a pedagogical introduction you may refer to)
7000:
6939:
6911:
6752:
6696:
1263:
are non-negative. If we define another orthonormal operator basis
7309:
Pearle, P. (2012). "Simple derivation of the Lindblad equation".
6028:
and no unitary evolution. In this case, the Lindblad equation is
1849:, is a linear superoperator. The semigroup can be recovered as
7293:
Quantum Mechanics of Non-Hamiltonian and Dissipative Systems
4256:{\displaystyle {\tilde {\chi }}(t)={\tilde {\rho }}(t)R_{0}}
3809:{\displaystyle {\dot {\tilde {\chi }}}=-i-\int _{0}^{t}dt']}
1332:
This reduces the master equation to the same form as before:
7228:
Attal, Stéphane; Joye, Alain; Pillet, Claude-Alain (2006).
6000:), above form of the Lindblad superoperator L is achieved.
4736:
If the interaction Hamiltonian is assumed to have the form
7066:
7295:. Amsterdam, Boston, London, New York: Elsevier Science.
3986:, of the aforementioned differo-integral equation yields
2360:
The Lindblad-type evolution of the density matrix in the
146:
The Lindblad master equation for system's density matrix
7003:"Relaxation of Quantum Systems with Equidistant Spectra"
6422:
of the photons, we can add a further unitary evolution:
7269:
7339:
7270:
Ingarden, Roman S.; Kossakowski, A.; Ohya, M. (1997).
6316:
6222:
6008:
In the simplest case, there is just one jump operator
4795:{\displaystyle H_{BS}=\sum _{i}\alpha _{i}\Gamma _{i}}
1114:
7133:
Accardi, Luigi; Lu, Yun Gang; Volovich, I.V. (2002).
6859:. Lecture Notes in Physics. Vol. 717. Springer.
6431:
6401:
6363:
6174:
6037:
6014:
5914:
5826:
5238:
4958:
4865:
4838:
4811:
4745:
4536:
4484:
4272:
4197:
3995:
3935:
3905:
3851:
3825:
3576:
3544:
3376:
3300:
3190:
3048:
2956:
2911:
2839:
2819:
2759:
2720:
2667:
2601:
2381:
2166:
2090:
1957:
1858:
1828:
1704:
1599:
1566:
1535:
1338:
1272:
1089:
1035:
1015:
988:
947:
760:
717:
670:
619:
541:
508:
471:
421:
395:
339:
159:
654:{\displaystyle {\dot {\rho }}={\mathcal {L}}(\rho )}
77:
describing open quantum systems. It generalizes the
7095:
7028:"On the generators of quantum dynamical semigroups"
6924:
6682:(2014). "Quantum Mechanics Without State Vectors".
3367:. This equation can be integrated directly to give
2806:{\displaystyle {\tilde {M}}=U_{0}MU_{0}^{\dagger }}
498:are a set of non-negative real coefficients called
6583:
6414:
6376:
6346:
6135:
6020:
5992:
5900:
5809:
5218:
4941:
4851:
4824:
4794:
4725:
4519:
4466:
4255:
4179:
3978:
3918:
3891:
3837:
3808:
3559:
3527:
3359:
3286:
3176:
3031:
2939:
2897:
2825:
2805:
2746:{\displaystyle \rho =\operatorname {tr} _{B}\chi }
2745:
2706:
2650:
2542:
2302:
2151:
2070:
1928:
1841:
1811:
1684:
1578:
1548:
1501:
1321:
1245:
1054:
1021:
1001:
966:
930:
727:
703:
653:
594:
527:
490:
447:
413:, describing the unitary aspects of the dynamics.
401:
375:
322:
7132:
6831:Lecture notes on Quantum Computation, Ph219/CS219
6790:
6653:
6628:SokhotskiâPlemelj theorem § Heitler function
3360:{\displaystyle U_{BS}=U_{0}^{\dagger }U(t,t_{0})}
7352:
2081:and also under the inhomogeneous transformation
751:More generally, the GKSL equation has the form
7230:Open Quantum Systems II: The Markovian Approach
7227:
3926:is the density operator of the bath initially.
7246:
1511:
36:GoriniâKossakowskiâSudarshanâLindblad equation
7336:Quantum jump (monte carlo) solver from QuTiP.
7211:Quantum Dynamical Semigroups and Applications
6857:Quantum Dynamical Semigroups and Applications
6654:Breuer, Heinz-Peter; Petruccione, F. (2002).
4520:{\displaystyle \rho (t')\rightarrow \rho (t)}
2152:{\displaystyle L_{i}\to L_{i}'=L_{i}+a_{i}I,}
5961:
5915:
5895:
5862:
5850:
5827:
5799:
5735:
5552:
5488:
5213:
5039:
4720:
4601:
4461:
4337:
4174:
4060:
3032:{\displaystyle {\dot {\tilde {\chi }}}=-i\,}
961:
948:
698:
686:
436:
422:
352:
340:
7208:
6971:
6854:
2561:, i.e. it preserves the identity operator.
6892:An Open Systems Approach to Quantum Optics
3929:Tracing over the bath degrees of freedom,
1322:{\displaystyle L_{i}=\sum _{j}u_{ji}A_{j}}
704:{\displaystyle {\dot {\rho }}=\{H,\rho \}}
595:{\displaystyle {\dot {\rho }}=-(i/\hbar )}
7165:
6938:
6910:
6805:10.1093/acprof:oso/9780199213900.001.0001
6751:
6695:
3028:
2898:{\displaystyle U_{0}=e^{i(H_{S}+H_{B})t}}
2647:
1695:The Lindblad equation can be obtained by
7025:
6678:
4527:on the right hand side of the equation.
1939:
83:trace-preserving and completely positive
7290:
7135:Quantum Theory and Its Stochastic Limit
6927:Open Systems & Information Dynamics
6737:
14:
7353:
7151:
6904:
3892:{\displaystyle \chi (0)=\rho (0)R_{0}}
2651:{\displaystyle H=H_{S}+H_{B}+H_{BS}\,}
2564:
609:The entire equation can be written in
535:one recovers the von Neumann equation
2364:can be equivalently described in the
2355:
1948:of Lindblad operators and constants,
711:. For this reason, the superoperator
7342:the quantum optics toolbox in Julia.
7209:Alicki, Robert; Lendi, Karl (1987).
6855:Alicki, Robert; Lendi, Karl (2007).
6827:
6733:
6731:
6649:
6647:
6645:
6643:
1069:is positive semidefinite, it can be
1560:indexed by a single time parameter
24:
6797:The Theory of Open Quantum Systems
6656:The Theory of Open Quantum Systems
6548:
6503:
6040:
5938:
5919:
5876:
5866:
5841:
5831:
5775:
5742:
5523:
5495:
5144:
5077:
4924:
4840:
4783:
1894:
1800:
1759:
1737:
1732:
1729:
1726:
1707:
720:
637:
25:
7377:
7321:
6897:
6728:
6640:
2707:{\displaystyle {\dot {\chi }}=-i}
2402:
1362:
784:
571:
491:{\displaystyle \gamma _{i}\geq 0}
183:
6903:This paragraph was adapted from
3560:{\displaystyle {\tilde {\chi }}}
141:
60:master equation in Lindblad form
6377:{\displaystyle {\overline {n}}}
1666:
746:
101:and are thus less general than
70:is one of the general forms of
6881:
6848:
6821:
6784:
6672:
6575:
6569:
6566:
6553:
6530:
6524:
6521:
6508:
6485:
6453:
6051:
6045:
5987:
5970:
5958:
5947:
5934:
5928:
5796:
5790:
5778:
5768:
5757:
5745:
5727:
5721:
5709:
5699:
5693:
5687:
5678:
5667:
5655:
5642:
5636:
5624:
5614:
5603:
5591:
5581:
5575:
5569:
5549:
5538:
5526:
5516:
5510:
5498:
5480:
5469:
5457:
5447:
5441:
5435:
5426:
5420:
5408:
5395:
5389:
5383:
5374:
5363:
5351:
5341:
5335:
5323:
5247:
5210:
5207:
5194:
5188:
5182:
5170:
5159:
5147:
5137:
5126:
5114:
5104:
5098:
5092:
5080:
5070:
5064:
5052:
5042:
4967:
4949:. The master equation becomes
4927:
4908:
4873:
4717:
4714:
4701:
4695:
4689:
4677:
4666:
4651:
4641:
4635:
4629:
4614:
4604:
4545:
4514:
4508:
4502:
4499:
4488:
4458:
4455:
4442:
4431:
4425:
4413:
4402:
4387:
4377:
4371:
4365:
4350:
4340:
4281:
4263:. The master equation becomes
4240:
4234:
4228:
4216:
4210:
4204:
4171:
4168:
4165:
4154:
4148:
4136:
4125:
4110:
4100:
4094:
4088:
4073:
4063:
4004:
3970:
3955:
3876:
3870:
3861:
3855:
3803:
3800:
3797:
3786:
3780:
3768:
3757:
3742:
3732:
3726:
3720:
3705:
3695:
3663:
3660:
3654:
3648:
3636:
3630:
3615:
3605:
3585:
3551:
3522:
3519:
3508:
3502:
3490:
3479:
3464:
3454:
3419:
3413:
3407:
3395:
3389:
3383:
3354:
3335:
3281:
3262:
3238:
3219:
3197:
3166:
3140:
3108:
3082:
3056:
3025:
3019:
2995:
2985:
2965:
2934:
2915:
2887:
2861:
2833:is an arbitrary operator, and
2766:
2701:
2689:
2419:
2407:
2170:
2101:
1982:
1920:
1914:
1881:
1875:
1795:
1789:
1773:
1767:
1743:
1718:
1712:
1660:
1654:
1632:
1629:
1623:
1610:
1379:
1367:
801:
789:
728:{\displaystyle {\mathcal {L}}}
661:which resembles the classical
648:
642:
589:
577:
574:
560:
200:
188:
13:
1:
7274:. New York: Springer Verlag.
7137:. New York: Springer Verlag.
6633:
2573:appearing in the equation is
528:{\displaystyle \gamma _{i}=0}
448:{\displaystyle \{L_{i}\}_{i}}
376:{\displaystyle \{a,b\}=ab+ba}
136:
112:
7346:The Lindblad master equation
7156:. Lecture Notes in Physics.
7118:10.1016/0550-3213(84)90184-6
6994:10.1016/0034-4877(72)90010-9
6369:
6332:
6243:
974:are arbitrary operators and
85:for any initial condition.
7:
7311:European Journal of Physics
7291:Tarasov, Vasily E. (2008).
7213:. Berlin: Springer Verlag.
6658:. Oxford University Press.
6601:
6415:{\displaystyle \omega _{c}}
6395:Hamiltonian with frequency
6393:quantum harmonic oscillator
6155:quantum harmonic oscillator
6146:This case is often used in
6003:
4852:{\displaystyle \Gamma _{i}}
4825:{\displaystyle \alpha _{i}}
3538:This implicit equation for
1822:which, by the linearity of
1524:quantum dynamical semigroup
1512:Quantum dynamical semigroup
461:phenomenologically modelled
10:
7382:
6714:10.1103/PhysRevA.90.042102
2940:{\displaystyle U(t,t_{0})}
1515:
97:, which can only describe
6957:10.1142/S1230161217400017
5229:which can be expanded as
1842:{\displaystyle \phi _{t}}
1549:{\displaystyle \phi _{t}}
967:{\displaystyle \{A_{m}\}}
741:Liouvillian superoperator
737:Lindbladian superoperator
7184:10.1007/3-540-46122-1_10
2580:A heuristic derivation,
2320:are complex numbers and
1518:Quantum Markov semigroup
7154:Dynamics of Dissipation
6894:. Springer Verlag, 1991
6608:Quantum master equation
6165:, with jump operators:
5820:The expectation values
1579:{\displaystyle t\geq 0}
1055:{\displaystyle N^{2}-1}
7328:Quantum Optics Toolbox
6793:Petruccione, Francesco
6585:
6416:
6378:
6348:
6137:
6022:
5994:
5902:
5811:
5220:
4943:
4853:
4826:
4796:
4727:
4521:
4468:
4257:
4181:
3980:
3920:
3893:
3839:
3810:
3561:
3529:
3361:
3288:
3178:
3042:where the Hamiltonian
3033:
2941:
2899:
2827:
2807:
2747:
2708:
2652:
2544:
2304:
2153:
2072:
1930:
1843:
1813:
1686:
1580:
1550:
1528:quantum dynamical maps
1503:
1396:
1323:
1256:where the eigenvalues
1247:
1075:unitary transformation
1056:
1023:
1003:
968:
932:
729:
705:
655:
596:
529:
492:
449:
403:
377:
324:
217:
7026:Lindblad, G. (1976).
6865:10.1007/3-540-70861-8
6791:Breuer, Heinz-Peter;
6598:propagation methods.
6586:
6417:
6379:
6349:
6157:(representing e.g. a
6138:
6023:
5995:
5903:
5812:
5221:
4944:
4854:
4827:
4805:for system operators
4797:
4728:
4522:
4469:
4258:
4182:
3981:
3921:
3919:{\displaystyle R_{0}}
3894:
3840:
3811:
3562:
3530:
3362:
3289:
3179:
3034:
2942:
2900:
2828:
2808:
2748:
2709:
2653:
2545:
2305:
2154:
2073:
1940:Invariance properties
1931:
1844:
1814:
1687:
1581:
1551:
1504:
1385:
1324:
1248:
1057:
1024:
1004:
1002:{\displaystyle A_{m}}
980:positive semidefinite
969:
933:
730:
706:
656:
597:
530:
493:
450:
404:
378:
325:
206:
105:, which can describe
6429:
6399:
6361:
6172:
6035:
6012:
5912:
5824:
5236:
4956:
4863:
4836:
4809:
4743:
4534:
4482:
4270:
4195:
3993:
3933:
3903:
3849:
3823:
3574:
3542:
3374:
3298:
3188:
3046:
2954:
2909:
2837:
2817:
2757:
2718:
2665:
2599:
2379:
2164:
2088:
1955:
1856:
1826:
1702:
1597:
1564:
1533:
1336:
1270:
1087:
1033:
1013:
986:
945:
758:
715:
668:
617:
539:
506:
469:
419:
393:
337:
157:
90:Schrödinger equation
79:Schrödinger equation
7176:2002LNP...597..239A
7110:1984NuPhB.244..125B
7081:1976JMP....17..821G
7044:1976CMaPh..48..119L
7019:1969JETP...29..145B
6986:1972RpMP....3..247K
6949:2017OSID...2440001C
6762:2020AIPA...10b5106M
6706:2014PhRvA..90d2102W
6623:Quantum jump method
6618:Open quantum system
6388:is the decay rate.
5294:
5014:
4832:and bath operators
4576:
4312:
4035:
3838:{\displaystyle t=0}
3683:
3442:
3331:
3261:
2802:
2565:Physical derivation
2510:
2464:
2362:Schrödinger picture
2282:
2244:
2116:
2014:
1999:
1526:—a family of
1472:
1439:
901:
868:
293:
260:
127:Schrödinger picture
99:pure quantum states
64:quantum Liouvillian
48:Andrzej Kossakowski
7052:10.1007/BF01608499
7032:Commun. Math. Phys
6888:Carmichael, Howard
6581:
6412:
6374:
6344:
6342:
6325:
6231:
6159:FabryâPerot cavity
6133:
6018:
5990:
5898:
5807:
5280:
5279:
5216:
5000:
4999:
4939:
4900:
4849:
4822:
4792:
4771:
4723:
4562:
4517:
4464:
4298:
4253:
4177:
4021:
3976:
3916:
3889:
3835:
3806:
3669:
3557:
3525:
3428:
3357:
3317:
3284:
3244:
3174:
3029:
2937:
2895:
2823:
2803:
2788:
2743:
2704:
2648:
2584:, in the notes by
2540:
2496:
2450:
2434:
2366:Heisenberg picture
2356:Heisenberg picture
2300:
2268:
2230:
2214:
2149:
2104:
2068:
2027:
2002:
1987:
1926:
1839:
1809:
1682:
1576:
1546:
1499:
1458:
1425:
1319:
1295:
1243:
1237:
1052:
1019:
999:
964:
928:
887:
854:
822:
725:
701:
663:Liouville equation
651:
611:superoperator form
604:Liouville equation
592:
525:
488:
445:
411:system Hamiltonian
399:
373:
320:
279:
246:
131:Heisenberg picture
7361:Quantum mechanics
7302:978-0-0805-5971-1
7281:978-0-7923-4473-5
7262:978-3-642-06094-6
7239:978-3-5403-0992-5
7220:978-0-3871-8276-6
7193:978-3-540-44111-3
7144:978-3-5404-1928-0
7098:Nuclear Physics B
6874:978-3-540-70860-5
6770:10.1063/1.5115323
6665:978-0-1985-2063-4
6613:Redfield equation
6441:
6372:
6335:
6324:
6246:
6230:
6086:
6021:{\displaystyle F}
5781:
5748:
5712:
5690:
5658:
5627:
5594:
5572:
5529:
5501:
5460:
5438:
5411:
5386:
5354:
5326:
5264:
5255:
5250:
5185:
5150:
5117:
5083:
5055:
4984:
4975:
4970:
4930:
4911:
4891:
4876:
4762:
4692:
4654:
4617:
4553:
4548:
4428:
4390:
4353:
4289:
4284:
4231:
4207:
4151:
4113:
4076:
4012:
4007:
3973:
3958:
3783:
3745:
3708:
3651:
3618:
3593:
3588:
3554:
3505:
3467:
3410:
3386:
3200:
3059:
3022:
2998:
2973:
2968:
2905:. Also note that
2826:{\displaystyle M}
2769:
2677:
2555:Ehrenfest theorem
2489:
2425:
2405:
2391:
2205:
2203:
2053:
2018:
2000:
1970:
1807:
1451:
1365:
1348:
1286:
1022:{\displaystyle N}
880:
807:
787:
770:
680:
629:
551:
402:{\displaystyle H}
272:
186:
169:
32:quantum mechanics
18:Lindblad equation
16:(Redirected from
7373:
7340:QuantumOptics.jl
7306:
7285:
7266:
7247:Gardiner, C.W.;
7243:
7224:
7205:
7169:
7167:quant-ph/0205188
7148:
7129:
7092:
7089:10.1063/1.522979
7063:
7022:
6997:
6968:
6942:
6917:
6916:
6914:
6901:
6895:
6885:
6879:
6878:
6852:
6846:
6845:
6843:
6837:. Archived from
6836:
6828:Preskill, John.
6825:
6819:
6818:
6788:
6782:
6781:
6755:
6735:
6726:
6725:
6699:
6680:Weinberg, Steven
6676:
6670:
6669:
6651:
6590:
6588:
6587:
6582:
6565:
6564:
6552:
6551:
6545:
6544:
6520:
6519:
6507:
6506:
6500:
6499:
6475:
6474:
6465:
6464:
6443:
6442:
6434:
6421:
6419:
6418:
6413:
6411:
6410:
6387:
6383:
6381:
6380:
6375:
6373:
6365:
6353:
6351:
6350:
6345:
6343:
6336:
6328:
6326:
6317:
6307:
6306:
6292:
6291:
6275:
6274:
6258:
6254:
6247:
6239:
6232:
6223:
6213:
6212:
6188:
6187:
6142:
6140:
6139:
6134:
6132:
6128:
6124:
6123:
6102:
6101:
6087:
6079:
6074:
6073:
6072:
6044:
6043:
6027:
6025:
6024:
6019:
5999:
5997:
5996:
5991:
5986:
5957:
5946:
5945:
5927:
5926:
5907:
5905:
5904:
5899:
5894:
5893:
5884:
5883:
5874:
5873:
5849:
5848:
5839:
5838:
5816:
5814:
5813:
5808:
5806:
5802:
5789:
5788:
5783:
5782:
5774:
5767:
5756:
5755:
5750:
5749:
5741:
5734:
5730:
5720:
5719:
5714:
5713:
5705:
5692:
5691:
5683:
5677:
5666:
5665:
5660:
5659:
5651:
5635:
5634:
5629:
5628:
5620:
5613:
5602:
5601:
5596:
5595:
5587:
5574:
5573:
5565:
5548:
5537:
5536:
5531:
5530:
5522:
5509:
5508:
5503:
5502:
5494:
5487:
5483:
5479:
5468:
5467:
5462:
5461:
5453:
5440:
5439:
5431:
5419:
5418:
5413:
5412:
5404:
5388:
5387:
5379:
5373:
5362:
5361:
5356:
5355:
5347:
5334:
5333:
5328:
5327:
5319:
5305:
5293:
5288:
5278:
5257:
5256:
5251:
5243:
5241:
5225:
5223:
5222:
5217:
5206:
5205:
5187:
5186:
5178:
5169:
5158:
5157:
5152:
5151:
5143:
5136:
5125:
5124:
5119:
5118:
5110:
5091:
5090:
5085:
5084:
5076:
5063:
5062:
5057:
5056:
5048:
5035:
5034:
5025:
5013:
5008:
4998:
4977:
4976:
4971:
4963:
4961:
4948:
4946:
4945:
4940:
4938:
4937:
4932:
4931:
4923:
4919:
4918:
4913:
4912:
4904:
4899:
4887:
4886:
4878:
4877:
4869:
4858:
4856:
4855:
4850:
4848:
4847:
4831:
4829:
4828:
4823:
4821:
4820:
4801:
4799:
4798:
4793:
4791:
4790:
4781:
4780:
4770:
4758:
4757:
4732:
4730:
4729:
4724:
4713:
4712:
4694:
4693:
4685:
4676:
4665:
4664:
4656:
4655:
4647:
4628:
4627:
4619:
4618:
4610:
4597:
4596:
4587:
4575:
4570:
4555:
4554:
4549:
4541:
4539:
4526:
4524:
4523:
4518:
4498:
4473:
4471:
4470:
4465:
4454:
4453:
4441:
4430:
4429:
4421:
4412:
4401:
4400:
4392:
4391:
4383:
4364:
4363:
4355:
4354:
4346:
4333:
4332:
4323:
4311:
4306:
4291:
4290:
4285:
4277:
4275:
4262:
4260:
4259:
4254:
4252:
4251:
4233:
4232:
4224:
4209:
4208:
4200:
4186:
4184:
4183:
4178:
4164:
4153:
4152:
4144:
4135:
4124:
4123:
4115:
4114:
4106:
4087:
4086:
4078:
4077:
4069:
4056:
4055:
4046:
4034:
4029:
4014:
4013:
4008:
4000:
3998:
3985:
3983:
3982:
3977:
3975:
3974:
3966:
3960:
3959:
3951:
3945:
3944:
3925:
3923:
3922:
3917:
3915:
3914:
3898:
3896:
3895:
3890:
3888:
3887:
3844:
3842:
3841:
3836:
3815:
3813:
3812:
3807:
3796:
3785:
3784:
3776:
3767:
3756:
3755:
3747:
3746:
3738:
3719:
3718:
3710:
3709:
3701:
3694:
3682:
3677:
3653:
3652:
3644:
3629:
3628:
3620:
3619:
3611:
3595:
3594:
3589:
3581:
3579:
3566:
3564:
3563:
3558:
3556:
3555:
3547:
3534:
3532:
3531:
3526:
3518:
3507:
3506:
3498:
3489:
3478:
3477:
3469:
3468:
3460:
3453:
3441:
3436:
3412:
3411:
3403:
3388:
3387:
3379:
3366:
3364:
3363:
3358:
3353:
3352:
3330:
3325:
3313:
3312:
3293:
3291:
3290:
3285:
3280:
3279:
3260:
3255:
3237:
3236:
3218:
3217:
3202:
3201:
3193:
3183:
3181:
3180:
3175:
3173:
3172:
3165:
3164:
3152:
3151:
3128:
3127:
3115:
3114:
3107:
3106:
3094:
3093:
3070:
3069:
3061:
3060:
3052:
3038:
3036:
3035:
3030:
3024:
3023:
3015:
3009:
3008:
3000:
2999:
2991:
2975:
2974:
2969:
2961:
2959:
2946:
2944:
2943:
2938:
2933:
2932:
2904:
2902:
2901:
2896:
2894:
2893:
2886:
2885:
2873:
2872:
2849:
2848:
2832:
2830:
2829:
2824:
2812:
2810:
2809:
2804:
2801:
2796:
2784:
2783:
2771:
2770:
2762:
2752:
2750:
2749:
2744:
2736:
2735:
2713:
2711:
2710:
2705:
2679:
2678:
2670:
2657:
2655:
2654:
2649:
2646:
2645:
2630:
2629:
2617:
2616:
2572:
2549:
2547:
2546:
2541:
2536:
2532:
2531:
2527:
2520:
2519:
2509:
2504:
2490:
2482:
2477:
2476:
2463:
2458:
2444:
2443:
2433:
2406:
2398:
2393:
2392:
2384:
2371:
2351:
2344:
2337:
2331:(unless all the
2330:
2323:
2319:
2309:
2307:
2306:
2301:
2287:
2283:
2281:
2276:
2267:
2266:
2254:
2253:
2243:
2238:
2224:
2223:
2213:
2204:
2202:
2191:
2180:
2158:
2156:
2155:
2150:
2142:
2141:
2129:
2128:
2112:
2100:
2099:
2077:
2075:
2074:
2069:
2064:
2063:
2054:
2052:
2051:
2042:
2040:
2039:
2026:
2010:
2001:
1995:
1986:
1981:
1980:
1971:
1969:
1968:
1959:
1947:
1935:
1933:
1932:
1927:
1913:
1912:
1903:
1902:
1898:
1897:
1874:
1873:
1848:
1846:
1845:
1840:
1838:
1837:
1818:
1816:
1815:
1810:
1808:
1806:
1798:
1788:
1787:
1766:
1765:
1752:
1750:
1749:
1735:
1711:
1710:
1691:
1689:
1688:
1683:
1653:
1652:
1622:
1621:
1609:
1608:
1585:
1583:
1582:
1577:
1558:density matrices
1556:on the space of
1555:
1553:
1552:
1547:
1545:
1544:
1508:
1506:
1505:
1500:
1498:
1494:
1493:
1489:
1482:
1481:
1471:
1466:
1452:
1444:
1438:
1433:
1421:
1420:
1406:
1405:
1395:
1393:
1366:
1358:
1350:
1349:
1341:
1328:
1326:
1325:
1320:
1318:
1317:
1308:
1307:
1294:
1282:
1281:
1262:
1252:
1250:
1249:
1244:
1242:
1241:
1234:
1233:
1226:
1225:
1160:
1159:
1126:
1125:
1099:
1098:
1079:
1068:
1061:
1059:
1058:
1053:
1045:
1044:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
998:
997:
977:
973:
971:
970:
965:
960:
959:
937:
935:
934:
929:
927:
923:
922:
918:
911:
910:
900:
895:
881:
873:
867:
862:
850:
849:
835:
834:
821:
788:
780:
772:
771:
763:
734:
732:
731:
726:
724:
723:
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682:
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673:
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641:
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631:
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553:
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531:
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497:
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489:
481:
480:
454:
452:
451:
446:
444:
443:
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433:
408:
406:
405:
400:
382:
380:
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374:
329:
327:
326:
321:
319:
315:
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310:
303:
302:
292:
287:
273:
265:
259:
254:
242:
241:
227:
226:
216:
214:
187:
179:
171:
170:
162:
149:
103:density matrices
75:master equations
52:George Sudarshan
21:
7381:
7380:
7376:
7375:
7374:
7372:
7371:
7370:
7351:
7350:
7324:
7303:
7282:
7263:
7240:
7221:
7194:
7145:
6974:Rep. Math. Phys
6921:
6920:
6902:
6898:
6886:
6882:
6875:
6853:
6849:
6841:
6834:
6826:
6822:
6815:
6799:. p. 125.
6789:
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6729:
6677:
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6666:
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6560:
6556:
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6515:
6511:
6502:
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6495:
6491:
6470:
6466:
6460:
6456:
6433:
6432:
6430:
6427:
6426:
6406:
6402:
6400:
6397:
6396:
6385:
6364:
6362:
6359:
6358:
6341:
6340:
6327:
6315:
6308:
6302:
6298:
6296:
6287:
6283:
6276:
6270:
6266:
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6262:
6238:
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6233:
6221:
6214:
6208:
6204:
6202:
6189:
6183:
6179:
6175:
6173:
6170:
6169:
6161:) coupled to a
6119:
6115:
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6092:
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6064:
6057:
6039:
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6036:
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4316:
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4079:
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3775:
3774:
3760:
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3678:
3673:
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3580:
3578:
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3497:
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3470:
3459:
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3401:
3378:
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3344:
3326:
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3301:
3299:
3296:
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3271:
3256:
3248:
3232:
3228:
3210:
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3160:
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3102:
3098:
3089:
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3078:
3074:
3062:
3051:
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3047:
3044:
3043:
3014:
3013:
3001:
2990:
2989:
2988:
2960:
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2928:
2924:
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2669:
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2600:
2597:
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2570:
2567:
2515:
2511:
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2500:
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2137:
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2108:
2095:
2091:
2089:
2086:
2085:
2059:
2055:
2047:
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2041:
2032:
2028:
2022:
2006:
1991:
1985:
1976:
1972:
1964:
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1893:
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1412:
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1407:
1401:
1397:
1394:
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1357:
1340:
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1337:
1334:
1333:
1313:
1309:
1300:
1296:
1290:
1277:
1273:
1271:
1268:
1267:
1261:
1257:
1236:
1235:
1221:
1217:
1216:
1212:
1210:
1205:
1200:
1194:
1193:
1188:
1183:
1178:
1172:
1171:
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1155:
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1149:
1143:
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1137:
1132:
1127:
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1084:
1077:
1066:
1040:
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1014:
1011:
1010:
993:
989:
987:
984:
983:
975:
955:
951:
946:
943:
942:
906:
902:
896:
891:
886:
882:
872:
863:
858:
845:
841:
840:
836:
827:
823:
811:
779:
762:
761:
759:
756:
755:
749:
719:
718:
716:
713:
712:
672:
671:
669:
666:
665:
636:
635:
621:
620:
618:
615:
614:
566:
543:
542:
540:
537:
536:
513:
509:
507:
504:
503:
476:
472:
470:
467:
466:
439:
435:
429:
425:
420:
417:
416:
394:
391:
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338:
335:
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264:
255:
250:
237:
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228:
222:
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178:
161:
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158:
155:
154:
147:
144:
139:
115:
44:Vittorio Gorini
28:
23:
22:
15:
12:
11:
5:
7379:
7369:
7368:
7363:
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7348:
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7331:
7323:
7322:External links
7320:
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7225:
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7104:(1): 125â134.
7093:
7064:
7023:
6998:
6969:
6919:
6918:
6896:
6880:
6873:
6847:
6844:on 2020-06-23.
6820:
6813:
6783:
6727:
6671:
6664:
6638:
6637:
6635:
6632:
6631:
6630:
6625:
6620:
6615:
6610:
6603:
6600:
6596:density matrix
6592:
6591:
6580:
6577:
6574:
6571:
6568:
6563:
6559:
6555:
6550:
6543:
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6186:
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6178:
6177:
6148:quantum optics
6144:
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6111:
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3481:
3476:
3473:
3466:
3463:
3456:
3452:
3449:
3445:
3440:
3435:
3431:
3427:
3424:
3421:
3418:
3415:
3409:
3406:
3400:
3397:
3394:
3391:
3385:
3382:
3356:
3351:
3347:
3343:
3340:
3337:
3334:
3329:
3324:
3320:
3316:
3311:
3308:
3304:
3283:
3278:
3274:
3270:
3267:
3264:
3259:
3254:
3251:
3247:
3243:
3240:
3235:
3231:
3227:
3224:
3221:
3216:
3213:
3209:
3205:
3199:
3196:
3171:
3168:
3163:
3159:
3155:
3150:
3146:
3142:
3139:
3136:
3132:
3126:
3123:
3119:
3113:
3110:
3105:
3101:
3097:
3092:
3088:
3084:
3081:
3077:
3073:
3068:
3065:
3058:
3055:
3040:
3039:
3027:
3021:
3018:
3012:
3007:
3004:
2997:
2994:
2987:
2984:
2981:
2978:
2972:
2967:
2964:
2936:
2931:
2927:
2923:
2920:
2917:
2914:
2892:
2889:
2884:
2880:
2876:
2871:
2867:
2863:
2860:
2856:
2852:
2847:
2843:
2822:
2800:
2795:
2791:
2787:
2782:
2778:
2774:
2768:
2765:
2742:
2739:
2734:
2730:
2726:
2723:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2676:
2673:
2659:
2658:
2644:
2641:
2637:
2633:
2628:
2624:
2620:
2615:
2611:
2607:
2604:
2566:
2563:
2551:
2550:
2539:
2535:
2530:
2526:
2523:
2518:
2514:
2508:
2503:
2499:
2494:
2488:
2485:
2480:
2475:
2471:
2467:
2462:
2457:
2453:
2448:
2442:
2438:
2432:
2428:
2424:
2421:
2418:
2415:
2412:
2409:
2404:
2401:
2396:
2390:
2387:
2357:
2354:
2348:
2341:
2334:
2327:
2316:
2311:
2310:
2299:
2296:
2293:
2290:
2286:
2280:
2275:
2271:
2265:
2261:
2257:
2252:
2248:
2242:
2237:
2233:
2228:
2222:
2218:
2212:
2208:
2201:
2198:
2194:
2189:
2186:
2183:
2179:
2176:
2172:
2169:
2159:
2148:
2145:
2140:
2136:
2132:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2098:
2094:
2079:
2078:
2067:
2062:
2058:
2050:
2046:
2038:
2035:
2031:
2025:
2021:
2017:
2013:
2009:
2005:
1998:
1994:
1990:
1984:
1979:
1975:
1967:
1963:
1941:
1938:
1937:
1936:
1925:
1922:
1919:
1916:
1911:
1907:
1901:
1896:
1890:
1886:
1883:
1880:
1877:
1872:
1869:
1866:
1862:
1836:
1832:
1820:
1819:
1805:
1802:
1797:
1794:
1791:
1786:
1782:
1778:
1775:
1772:
1769:
1764:
1761:
1757:
1748:
1745:
1742:
1739:
1734:
1731:
1728:
1723:
1720:
1717:
1714:
1709:
1693:
1692:
1681:
1678:
1675:
1672:
1669:
1665:
1662:
1659:
1656:
1651:
1648:
1645:
1641:
1637:
1634:
1631:
1628:
1625:
1620:
1616:
1612:
1607:
1603:
1586:that obey the
1575:
1572:
1569:
1543:
1539:
1516:Main article:
1513:
1510:
1497:
1492:
1488:
1485:
1480:
1476:
1470:
1465:
1461:
1456:
1450:
1447:
1442:
1437:
1432:
1428:
1424:
1419:
1415:
1410:
1404:
1400:
1392:
1388:
1384:
1381:
1378:
1375:
1372:
1369:
1364:
1361:
1356:
1353:
1347:
1344:
1330:
1329:
1316:
1312:
1306:
1303:
1299:
1293:
1289:
1285:
1280:
1276:
1259:
1254:
1253:
1240:
1232:
1229:
1224:
1220:
1215:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1195:
1192:
1189:
1187:
1184:
1182:
1179:
1177:
1174:
1173:
1170:
1167:
1165:
1162:
1158:
1154:
1150:
1148:
1145:
1144:
1141:
1138:
1136:
1133:
1131:
1128:
1124:
1120:
1116:
1115:
1113:
1108:
1105:
1102:
1097:
1093:
1051:
1048:
1043:
1039:
1018:
996:
992:
963:
958:
954:
950:
939:
938:
926:
921:
917:
914:
909:
905:
899:
894:
890:
885:
879:
876:
871:
866:
861:
857:
853:
848:
844:
839:
833:
830:
826:
820:
817:
814:
810:
806:
803:
800:
797:
794:
791:
786:
783:
778:
775:
769:
766:
748:
745:
735:is called the
722:
700:
697:
694:
691:
688:
685:
679:
676:
650:
647:
644:
639:
634:
628:
625:
591:
588:
585:
582:
579:
576:
573:
569:
565:
562:
559:
556:
550:
547:
524:
521:
516:
512:
487:
484:
479:
475:
457:jump operators
442:
438:
432:
428:
424:
398:
385:anticommutator
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
331:
330:
318:
313:
309:
306:
301:
297:
291:
286:
282:
277:
271:
268:
263:
258:
253:
249:
245:
240:
236:
231:
225:
221:
213:
209:
205:
202:
199:
196:
193:
190:
185:
182:
177:
174:
168:
165:
143:
140:
138:
135:
123:density matrix
114:
111:
56:Göran Lindblad
42:, named after
26:
9:
6:
4:
3:
2:
7378:
7367:
7364:
7362:
7359:
7358:
7356:
7347:
7344:
7341:
7338:
7335:
7332:
7329:
7326:
7325:
7316:
7312:
7308:
7304:
7298:
7294:
7289:
7288:
7283:
7277:
7273:
7268:
7264:
7258:
7254:
7253:Quantum Noise
7250:
7249:Zoller, Peter
7245:
7241:
7235:
7231:
7226:
7222:
7216:
7212:
7207:
7203:
7199:
7195:
7189:
7185:
7181:
7177:
7173:
7168:
7163:
7159:
7155:
7150:
7146:
7140:
7136:
7131:
7127:
7123:
7119:
7115:
7111:
7107:
7103:
7099:
7094:
7090:
7086:
7082:
7078:
7074:
7070:
7069:J. Math. Phys
7065:
7061:
7057:
7053:
7049:
7045:
7041:
7037:
7033:
7029:
7024:
7020:
7016:
7012:
7008:
7004:
6999:
6995:
6991:
6987:
6983:
6979:
6975:
6970:
6966:
6962:
6958:
6954:
6950:
6946:
6941:
6936:
6932:
6928:
6923:
6922:
6913:
6908:
6900:
6893:
6889:
6884:
6876:
6870:
6866:
6862:
6858:
6851:
6840:
6833:
6832:
6824:
6816:
6814:9780199213900
6810:
6806:
6802:
6798:
6794:
6787:
6779:
6775:
6771:
6767:
6763:
6759:
6754:
6749:
6746:(2): 025106.
6745:
6741:
6734:
6732:
6723:
6719:
6715:
6711:
6707:
6703:
6698:
6693:
6690:(4): 042102.
6689:
6685:
6681:
6675:
6667:
6661:
6657:
6650:
6648:
6646:
6644:
6639:
6629:
6626:
6624:
6621:
6619:
6616:
6614:
6611:
6609:
6606:
6605:
6599:
6597:
6578:
6572:
6561:
6557:
6541:
6537:
6533:
6527:
6516:
6512:
6496:
6492:
6488:
6482:
6479:
6476:
6471:
6467:
6461:
6457:
6450:
6447:
6444:
6438:
6435:
6425:
6424:
6423:
6407:
6403:
6394:
6391:To model the
6389:
6366:
6337:
6329:
6321:
6318:
6312:
6310:
6303:
6299:
6293:
6288:
6284:
6280:
6278:
6271:
6267:
6259:
6255:
6251:
6248:
6240:
6234:
6227:
6224:
6218:
6216:
6209:
6205:
6199:
6196:
6193:
6191:
6184:
6180:
6168:
6167:
6166:
6164:
6160:
6156:
6151:
6149:
6129:
6125:
6120:
6116:
6112:
6109:
6106:
6103:
6098:
6094:
6089:
6083:
6080:
6075:
6069:
6065:
6061:
6058:
6054:
6048:
6031:
6030:
6029:
6015:
6001:
5983:
5980:
5976:
5973:
5967:
5964:
5954:
5951:
5942:
5931:
5923:
5890:
5886:
5880:
5870:
5859:
5856:
5853:
5845:
5835:
5803:
5793:
5785:
5764:
5761:
5752:
5731:
5724:
5716:
5706:
5696:
5684:
5674:
5671:
5662:
5652:
5645:
5639:
5631:
5621:
5610:
5607:
5598:
5588:
5578:
5566:
5559:
5555:
5545:
5542:
5533:
5513:
5505:
5484:
5476:
5473:
5464:
5454:
5444:
5432:
5423:
5415:
5405:
5398:
5392:
5380:
5370:
5367:
5358:
5348:
5338:
5330:
5320:
5312:
5307:
5302:
5299:
5295:
5290:
5285:
5281:
5275:
5272:
5269:
5265:
5261:
5258:
5252:
5244:
5232:
5231:
5230:
5202:
5198:
5191:
5179:
5173:
5166:
5163:
5154:
5133:
5130:
5121:
5111:
5101:
5095:
5087:
5067:
5059:
5049:
5036:
5031:
5027:
5022:
5019:
5015:
5010:
5005:
5001:
4995:
4992:
4989:
4985:
4981:
4978:
4972:
4964:
4952:
4951:
4950:
4934:
4915:
4905:
4896:
4892:
4888:
4883:
4880:
4870:
4844:
4817:
4813:
4787:
4777:
4773:
4767:
4763:
4759:
4754:
4751:
4747:
4739:
4738:
4737:
4709:
4705:
4698:
4686:
4680:
4673:
4670:
4661:
4658:
4648:
4638:
4632:
4624:
4621:
4611:
4598:
4593:
4589:
4584:
4581:
4577:
4572:
4567:
4563:
4559:
4556:
4550:
4542:
4530:
4529:
4528:
4511:
4505:
4495:
4492:
4485:
4450:
4446:
4438:
4435:
4422:
4416:
4409:
4406:
4397:
4394:
4384:
4374:
4368:
4360:
4357:
4347:
4334:
4329:
4325:
4320:
4317:
4313:
4308:
4303:
4299:
4295:
4292:
4286:
4278:
4266:
4265:
4264:
4248:
4244:
4237:
4225:
4219:
4213:
4201:
4161:
4158:
4145:
4139:
4132:
4129:
4120:
4117:
4107:
4097:
4091:
4083:
4080:
4070:
4057:
4052:
4048:
4043:
4040:
4036:
4031:
4026:
4022:
4018:
4015:
4009:
4001:
3989:
3988:
3987:
3967:
3961:
3952:
3946:
3941:
3937:
3927:
3911:
3907:
3884:
3880:
3873:
3867:
3864:
3858:
3852:
3832:
3829:
3826:
3793:
3790:
3777:
3771:
3764:
3761:
3752:
3749:
3739:
3729:
3723:
3715:
3712:
3702:
3691:
3688:
3684:
3679:
3674:
3670:
3666:
3657:
3645:
3639:
3633:
3625:
3622:
3612:
3602:
3599:
3596:
3590:
3582:
3570:
3569:
3568:
3548:
3515:
3512:
3499:
3493:
3486:
3483:
3474:
3471:
3461:
3450:
3447:
3443:
3438:
3433:
3429:
3425:
3422:
3416:
3404:
3398:
3392:
3380:
3370:
3369:
3368:
3349:
3345:
3341:
3338:
3332:
3327:
3322:
3318:
3314:
3309:
3306:
3302:
3276:
3272:
3268:
3265:
3257:
3252:
3249:
3245:
3241:
3233:
3229:
3225:
3222:
3214:
3211:
3207:
3203:
3194:
3169:
3161:
3157:
3153:
3148:
3144:
3137:
3134:
3130:
3124:
3121:
3117:
3111:
3103:
3099:
3095:
3090:
3086:
3079:
3075:
3071:
3066:
3063:
3053:
3016:
3010:
3005:
3002:
2992:
2982:
2979:
2976:
2970:
2962:
2950:
2949:
2948:
2929:
2925:
2921:
2918:
2912:
2890:
2882:
2878:
2874:
2869:
2865:
2858:
2854:
2850:
2845:
2841:
2820:
2798:
2793:
2789:
2785:
2780:
2776:
2772:
2763:
2740:
2737:
2732:
2728:
2724:
2721:
2698:
2695:
2692:
2686:
2683:
2680:
2674:
2671:
2642:
2639:
2635:
2631:
2626:
2622:
2618:
2613:
2609:
2605:
2602:
2595:
2594:
2593:
2589:
2587:
2583:
2578:
2576:
2562:
2560:
2556:
2537:
2533:
2528:
2524:
2521:
2516:
2512:
2506:
2501:
2497:
2492:
2486:
2483:
2478:
2473:
2469:
2465:
2460:
2455:
2451:
2446:
2440:
2436:
2430:
2426:
2422:
2416:
2413:
2410:
2399:
2394:
2388:
2385:
2375:
2374:
2373:
2367:
2363:
2353:
2297:
2294:
2291:
2288:
2284:
2278:
2273:
2269:
2263:
2259:
2255:
2250:
2246:
2240:
2235:
2231:
2226:
2220:
2216:
2210:
2206:
2199:
2196:
2192:
2187:
2184:
2181:
2177:
2174:
2167:
2160:
2146:
2143:
2138:
2134:
2130:
2125:
2121:
2117:
2113:
2109:
2105:
2096:
2092:
2084:
2083:
2082:
2065:
2060:
2056:
2048:
2044:
2036:
2033:
2029:
2023:
2019:
2015:
2011:
2007:
2003:
1996:
1992:
1988:
1977:
1973:
1965:
1961:
1951:
1950:
1949:
1923:
1917:
1909:
1905:
1899:
1888:
1884:
1878:
1870:
1867:
1864:
1860:
1852:
1851:
1850:
1834:
1830:
1803:
1792:
1784:
1780:
1776:
1770:
1762:
1755:
1746:
1740:
1721:
1715:
1698:
1697:
1696:
1679:
1676:
1673:
1670:
1667:
1663:
1657:
1649:
1646:
1643:
1639:
1635:
1626:
1618:
1614:
1605:
1601:
1593:
1592:
1591:
1589:
1573:
1570:
1567:
1559:
1541:
1537:
1529:
1525:
1519:
1509:
1495:
1490:
1486:
1483:
1478:
1474:
1468:
1463:
1459:
1454:
1448:
1445:
1440:
1435:
1430:
1426:
1422:
1417:
1413:
1408:
1402:
1398:
1390:
1386:
1382:
1376:
1373:
1370:
1359:
1354:
1351:
1345:
1342:
1314:
1310:
1304:
1301:
1297:
1291:
1287:
1283:
1278:
1274:
1266:
1265:
1264:
1238:
1230:
1227:
1222:
1218:
1213:
1207:
1202:
1197:
1190:
1185:
1180:
1175:
1168:
1163:
1156:
1152:
1146:
1139:
1134:
1129:
1122:
1118:
1111:
1106:
1103:
1100:
1095:
1091:
1083:
1082:
1081:
1076:
1072:
1063:
1049:
1046:
1041:
1037:
1016:
994:
990:
981:
956:
952:
924:
919:
915:
912:
907:
903:
897:
892:
888:
883:
877:
874:
869:
864:
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586:
583:
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548:
545:
522:
519:
514:
510:
501:
500:damping rates
485:
482:
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464:
462:
458:
455:are a set of
440:
430:
426:
414:
412:
396:
388:
386:
370:
367:
364:
361:
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346:
343:
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284:
280:
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251:
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211:
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180:
175:
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153:
152:
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142:Diagonal form
134:
132:
128:
124:
119:
110:
108:
104:
100:
96:
95:state vectors
91:
86:
84:
80:
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7232:. Springer.
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6839:the original
6830:
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6740:AIP Advances
6739:
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6684:Phys. Rev. A
6683:
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740:
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120:
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107:mixed states
87:
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68:Lindbladian
7355:Categories
7330:for Matlab
7075:(5): 821.
7038:(2): 119.
6980:(4): 247.
6940:1710.05993
6912:1802.00010
6753:1906.04478
6634:References
137:Definition
113:Motivation
7366:Equations
7317:(4), 805.
7202:118089738
6778:184487806
6697:1405.3483
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7060:55220796
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6722:53990012
6602:See also
6004:Examples
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1491:}
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1123:1
1112:[
1107:=
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920:}
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189:[
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