Quantum Dynamics in Classical Spin Baths Alessandro Sergi∗ School of Chemistry and Physics, University of KwaZulu-Natal, Pietermaritzburg, Private Bag X01 Scottsville, 3209 Pietermaritzburg
arXiv:1306.3299v1 [quant-ph] 14 Jun 2013
and National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, South Africa
Abstract A formalism for studying the dynamics of quantum systems embedded in classical spin baths is introduced. The theory is based on generalized antisymmetric brackets and predicts the presence of open-path off-diagonal geometric phases in the evolution of the density matrix. The weak coupling limit of the equation can be integrated by standard algorithms and provides a non-Markovian approach to the computer simulation of quantum systems in classical spin environments. It is expected that the theory and numerical schemes presented here have a wide applicability.
∗
[email protected]
1
In recent years, dissipation and decoherence in quantum systems have come to be considered more as resources to be exploited [1–3] than as simple nuisances to be avoided at all cost. The theoretical study of both effects requires to couple the system of interest to an environment [4]. This latter can be represented either by means of bosonic degrees of freedom [5] or by spinors [6–8]. Full quantum theories of both system and environment are difficult to simulate and usually one resorts to approached based on master equations [4, 9–13]. However, there are many situations where the environment’s coordinates can be considered to follow the laws of classical mechanics. When the such coordinates are canonically conjugate momenta and positions, one approach to treat these situations is provided by the quantumclassical Liouville equation [14–28]. Nevertheless, there are also many interesting cases when the bath is described in terms of classical spins, e.g., in order to model complex molecules where magnetic effects are important [29, 30]. Such models can be studied by means of Monte Carlo methods or by Molecular Dynamics simulations [31, 32]. The scope of this communication is to generalize the quantum-classical Liouville equation [14–28] in order to study the dynamics of quantum systems embedded in baths of classical spins. This is naturally achieved within a quantum-classical theory formulated by means of generalized antisymmetric brackets [33–35]. It is worth noting that antisymmetric brackets have also been used to formulate the statistical mechanics of classical systems with thermodynamic [36–38] and holonomic constraints [39, 40]. The quantum-classical Liouville equation for spin baths must be represented in some basis to be of practical use and, in the regime of weak coupling, the adiabatic basis is a good choice. However, reasonable forms of the coupling between the quantum subsystem and the classical spin bath cause such a basis to be complex (whereas, when the bath is described by canonical variables, the adiabatic basis is real in the absence of magnetic fields). Hence, an interesting prediction of the quantum-classical Liouville equation for spin baths is the presence of open-path [41] geometric phases [42–44] in the evolution of the off-diagonal elements [45–47] of the density matrix. In order to introduce the formalism, one can consider a classical spin vector S whose energy is described by the Hamiltonian HSB (S), and denote its components with SI , I = x, y, z. The generalization to systems with many spins is straightforward. The equations of 2
motion can be written in matrix form as S˙ I =
X
S BIJ
J
∂HSB , ∂SJ
(1)
where
Sz −Sy 0 S B = Sx . −Sz 0 Sy −Sx
0
(2)
S The antisymmetric matrix BS can also be written in a compact way as BIJ = ǫIJK SK ,
where ǫIJK is the completely antisymmetric tensor (or Levi-Civita symbol). The equations of motion (1) preserve the Casimir C2 = S · S for any arbitrary Hamiltonian HSB (S). They P also have have zero phase space compressibility κ = I ∂ S˙ I /∂SI = 0. The classical equations
of motion of the spin can be written upon introducing the non-canonical bracket {A, B} =
X I,J
∂A S ∂B B , ∂SI IJ ∂SJ
(3)
where A = A(S) and B = B(S) are arbitrary functions of the spin degrees of freedom, in the form S˙ I = {SI , HSB } .
(4)
At this stage, the quantum-classical theory can be introduced assuming that the classical spin ˆ χ}) system interacts with a quantum system with a Hamiltonian H({ ˆ through an interaction ˆ c ({χ}, of the form H ˆ S) Hence, the total Hamiltonian operator of the quantum subsystem in the classical spin bath can be written as ˆ ˆ χ}) ˆ C ({χ}, H(S) = H({ ˆ +H ˆ S) + HSB (S) .
(5)
Given the quantum-classical density matrix ρˆ(S, t), the key step is postulating, in analogy with the case of the quantum-classical Liouville equation for canonical coordinates, the evolution of of the quantum system in the classical spin bath by means of the equation
ˆ H(S) i ∂ S ˆ ρˆ(S, t) = − , H(S) ρˆ(S, t) · D · ∂t h ¯ ρˆ(S, t)
=−
i ihˆ H(S), ρˆ(S, t) S D h ¯
3
(6)
where DS =
0 −1 −
i¯ h 2
P
→ − ← − ∂ BS ∂
I,J ∂SI
IJ ∂SJ
1+
i¯ h 2
P
→ − ← − ∂ BS ∂
I,J ∂SI
0
IJ ∂SJ
. (7)
Equation (6) is the quantum-classical Liouville equation for quantum systems in classical spin baths. Its proposal is the main result of this work. The right hand side of Eq. (6) defines what, in the case of canonically conjugate phase space coordinates, is known as quantum-classical bracket [14–16] or non-Hamiltonian commutator [33]. The bracket couples the dynamics of the classical degrees of freedom with that of the quantum operators taking into account both the conservation of the energy and ˆ C = 0, the bracket the quantum back-reaction. Moreover, when there is no coupling, i.e., H makes the quantum system evolve in terms of the standard commutator and the classical bath through the Poisson bracket, which in the present case has the non-canonical form given in Eq. (3). The quantum-classical spin bracket in Eq. (6) does not satisfy the Jacobi relation as its canonical coordinate counterpart [14–16] and this, in turn, leads to the lack of time-translation invariance of the algebra defined in terms of the bracket itself [33, 48]. Less abstract consequence of such mathematical features are that the coordinates of the spin bath, which are classical at time zero, acquire quantum phases during the dynamics. In practice, fast bath decoherence alleviates this problem and, indeed, the canonical version of the quantum-classical bracket, or of the quantum-classical Liouville equation, is used for many applications in chemistry and physics [14–28]. Moreover, it is worth reminding that the non-Lie (or, as they are also called, non-Hamiltonian) brackets, with their lack of time translation invariance, are also used to impose thermodynamical (such as constant temperature and/or pressure) [36–38] and holonomic constraints [39, 40] in classical molecular dynamics simulations [31, 32]. In order to represent the abstract Eq. (6) the quantum-classical Hamiltonian of Eq. (5) can be split as ˆ ˆ H(S) = HSB (S) + h(S) .
(8)
Accordingly, the adiabatic basis is defined by the eigenvalue equation ˆ h(S)|α; Si = Eα (S)|α; Si . 4
(9)
As one can see, the adiabatic basis in spin baths depends on all the non-canonical spin coordinates S, while in the canonical case it depends only on the positions R and not on the conjugate momenta P . In such a basis, Eq. (6) becomes ∂t ραα′ = −iωαα′ ραα′ −
X
S BIJ
I,J
"
∂HSB ∂ ρˆ ′ hα| |α i ∂SJ ∂SI
ˆ ∂ ρˆ ˆ ∂h 1 ∂ ρˆ ∂ h 1 |α′ i − hα| |α′i + hα| 2 ∂SI ∂SJ 2 ∂SI ∂SJ
#
(10) where we have used the antisymmetry of BS . Defining the coupling vector → − ∂ I dσα = hσ| |αi ∂SI
(11)
one finds the two identities hα|∂I ρˆ|α′i = ∂I ραα′ + dIασ ρσα′ − ρασ′ dIσ′ α′ ˆ ∂h ∂hασ hα| |σi = − ∆Eασ dIασ ∂SI ∂SI
(12) (13)
where ∆Eασ = Eα −Eσ . With the help of Eqs. (12) and (13), the quantum-classical Liouville equation for spin baths is written as ∂t ραα′ =
X
(−iωαα′ δαβ δαα′ − Lαα′ δαβ δαα′ − Jαα′ ,ββ ′
ββ ′
+ Sαα′ ,ββ ′ ) ρββ ′ .
(14)
In Eq. (14) the classical-like spin-Liouville operator Lαα′ =
X
S BIJ
I,J
1 ∂Eα′ ∂ ∂HSB ∂ + ∂SJ ∂SI 2 ∂SJ ∂SI
1 ∂Eα ∂ + 2 ∂SJ ∂SI
!
=
X
S BIJ
I,J
S ∂ ∂Hαα ′ ∂SJ ∂SI
(15)
S has been defined. The symbol Hαα ′ denotes the average adiabatic surface Hamiltonian S Hαα ′ = HSB +
1 (Eα + Eα′ ) . 2
(16)
The transition operator for the spin bath is given by Jαα′ ,ββ ′ =
X I,J
S BIJ
"
1 ∂ ∂HSB I dαβ δβ ′ α′ + ∆Eαβ dIαβ δα′ β ′ ∂SJ 2 ∂SJ
∂ 1 ∂HSB I∗ dα′ β ′ δαβ + ∆Eα′ β ′ dI∗ δαβ + α′ β ′ ∂SJ 2 ∂SJ
#
. (17)
5
The operator in Eq. (17) goes to the usual jump operator [23] when canonical variables are considered. For the spin bath a higher order transition operator must be introduced Sαα′ ,ββ ′ =
X
S BIJ
I,J
"
1 ∂(Eα + Eα′ ) J dαβ δα′ β ′ 2 ∂SI
1 ∂(Eα + Eα′ ) J∗ dα′ β ′ δαβ 2 ∂SI 1 1 − ∆Eασ dIασ dJσβ δα′ β ′ − ∆Eαβ dIαβ dJ∗ α′ β ′ 2 2 1 J∗ − ∆Eα′ σ′ dI∗ α′ σ′ dσ′ β ′ δαβ 2 1 J d − ∆Eα′ β ′ dI∗ ′ ′ α β αβ . 2 +
(18)
The operator in Eq. (18) is identically zero for canonical conjugate variables. The general equation of motion (14) is difficult to integrate. However, one can consider its weak-coupling limit upon taking the adiabatic limit of the operators in Eqs. (17) and (18). This is performed considering the off-diagonal elements of dαα′ , which couples different adiabatic energy surfaces, to be negligible. In such a limit, one obtains ad Jαα ′ ,ββ ′ = −
X
S BIJ
I,J
= −i
X
∂HSB I dαα + dI∗ α′ α′ δαβ δβ ′ α′ ∂SJ
S BIJ
I,J
∂HSB I φαα − φIα′ α′ δαβ δβ ′ α′ ∂SJ
(19)
where using the fact that dIαα is purely imaginary a phase φIαα = −idIαα
(20)
can be introduced. In a similar way one can take the adiabatic limit of the Sαα′ ,ββ ′ in Eq. (18) ad Sαα ′ ,ββ ′ = −
i X S ∂(Eα + Eα′ ) I φαα − φIα′ α′ δαα δα′ α′ BIJ 2 I,J ∂SJ
(21)
Hence, the weak-coupling (adiabatic) equation of motion reads
S X ∂Hαα ∂ ′ ′ ′ φIαα − φIα′ α′ ραα = −iωαα − i BIJ ∂t ∂SJ I,J
−
X
K,L
BKL
S ∂Hαα ′
∂SL 6
∂ ραα′ ∂SK
(22)
In Eq. (22) one can note the presence of two phase terms. While the phase ωαα′ is dynamical, the phase φαα is of a geometric origin and it is analogous to the famous Berry phase [42–44]. It is predicted by Eq. (6) to be present also for open paths [41] of the classical environment and it is off-diagonal in nature [45–47] since it can only be different from zero for the offdiagonal elements of ραα′ . It easy to see that the geometric phase φαα can be obtained without invoking the adiabatic limit upon selecting the diagonal elements dIαα′ in Eq. (14). Hence, it is remarkable that the geometric phase φαα is predicted also for non-adiabatic dynamics. The prediction of the geometric phase in Eq. (20) is the second important result of this work. In the absence of explicit time-dependences, Eq. (22) can be rewritten as "
d d − hα, S| |α, Si − hα′ , S| |α′ , Si dt dt
∂t ραα′ = −iωαα′ −
X I,J
!
S S ∂Hαα′ ∂ BIJ ραα′ ∂SJ ∂SI
(23)
Using Dyson identity, this can be written in propagator form as
ραα′ (t) = exp −i "
× exp −
Z
Z
t
t0 t
t0
′
′
dt ωαα′ (t ) ′
dt
× exp −(t − t0 )
d d hα, S| ′ |α, Si − hα′ , S| ′ |α′ , Si dt dt X I,J
S S ∂Hαα′ ∂ BIJ ραα′ (t0 ) ∂SJ ∂SI
!#
(24)
Equation (24) is the adiabatic approximation of the quantum-classical Liouville equation for spin baths, whose general form is given in Eq. (14). The propagator form of Eq. (24) is suitable to the development of an effective numerical scheme: as in the case of baths of canonically conjugate phase space coordinates, the evolution of the quantum system can be represented in term of the propagation of classical spin trajectories over coupled adiabatic energy surfaces. Such a scheme is complementary to those based on the solutions of master equations [4, 9–13]. However, it does not require to approximate in any form the memory effects of the bath since its degrees of freedom are described explicitly, in the spirit of molecular dynamics simulations [31, 32]. From this point of view, the approach presented here provides a non-Markovian route to the simulation of quantum effects in classical spin baths. It is expected that the theory and numerical schemes presented here have a wide applicability [6–8, 29, 30]. 7
This work is based upon research supported by the National Research Foundation of South Africa.
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