sign in sign up
<<
Home , Density matrix, Operator (physics)

of the Reduced Density

Nicholas Wheeler, Reed College Physics Department Spring 2009

Introduction. mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The properties of the system are latent then in the reduced density operator

ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ.

Concerning the specific meaning of (1). Let n) be an orthonormal | in the H of the (open) system, and N) be an s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗

tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e

The dynamics of the composite system is generated by Hamiltonian of the form

H = H s + H e + H i 2 Motion of the reduced density operator where

H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent.

Interaction picture. In the Schr¨odinger picture one has (on the assumption that the Hamiltonian is time-independent)

(i/!) H t i! ∂t ψ = H ψ = ψ = U(t) ψ with U(t) = e− | %t | %t ⇒ | %t | %0 Looking to the quantum motion of the expectation value of a time-independent observable A, we in the Schr¨odinger picture have

A = ψ A ψ = ψ U+(t)A U(t) ψ $ %t t$ | | %t 0$ | | %0 which in the becomes

= ψ A ψ with A U+(t)A U(t) 0$ | t| %0 t ≡ 0 giving i! ∂t A = [A, H] Compare this with the equation that in the Schr¨odinger picture is satisfied by the density ρ ψ ψ : ≡ | %$ |

i! ∂tρ = [H, ρ]

+ which entails ρt = U(t)ρ0 U (t). We have

tr ρ A in the Schr¨odinger picture A = t 0 $ %t tr ρ A in the Heisenberg picture ( ! 0 t" which shows very clearly the! distinction" between and equivalence of the two pictures. Suppose now that the Hamiltonian can be resolved into the sum

H = H 0 + H 1 3 of an “easy/uninteresting part” H 0 and a relatively “difficult/interesting part” i H 1. In the absence of H 1 we would have ψ ψ = exp H 0t ψ so 0 t ! 0 the time-dependent unitary transformation| % −→ | % − | % ) * i ψ Ψ = exp + H 0t ψ | %t −→ | %t ! | %t produces a state vector that in the absence)of H 1 would* not move at all. In the presence of H 1 we have

i i! ∂t Ψ t = exp + H 0t H 0 ψ + i! ∂t ψ t | % ! − | %t | % ) i *$ ' = exp + H 0t H 0 ψ + H 0 + H 1 ψ t ! − | %t | % –1 $ 'i = U ()t)H 1 U 0(*t) Ψ t with %U 0(t) exp& H 0t 0 | % ≡ − ! G(t) Ψ (2) ≡ | %t ) * where the hermiticity of the time-dependent operator G(t) is manifest, as is the fact that G(t) I in cases where H 0. We now have → 1 → A = ψ A ψ = Ψ B(t) Ψ $ %t t$ | | %t t$ | | %t where B(t) U –1(t)A U (t) ≡ 0 0 executes simple H 1-independent Heisenberg motion:

i! ∂t B(t) = [B(t), H 0] The solution of (2) is famously difficult to construct (involves chronological ordering), but has necessarily the form Ψ = V(t) Ψ : V(t) is unitary | %t | %0 ↓ = Ψ for all t in cases where H vanishes | %0 1 From ψ t = U 0(t) Ψ t = U 0(t)V(t) Ψ 0 = U 0(t)V(t) ψ 0 = U(t) ψ 0 we see | % | % | % i | % | % that we have in effect factored U(t) = exp (H 0 + H 1)t , and have thus ! placed ourselves in position to write − ) * A = tr ρ A = tr U (t)V(t)ρ V –1(t)U –1(t)A $ %t t 0 0 0 0 0 = tr V(t)ρ V –1(t) U –1(t)A U (t) ! " ! 0 · 0 0 0 " in which ρ is propelled (prograde)!by " +(i/ ) H t (i/ )( H + H ) t V(t) = e ! 0 e− ! 0 1 (3) · (i/ ) H 0 t while A is propelled (retrograde) by U 0(t) = e− ! . The expression on the right hand side of (3) is easy to write down but in typical cases is difficult to evaluate explicitly. In favorable cases one can appeal to Campbell-Baker-Hausdorff theory, but more commonly one works from

–1 i! ∂t V(t) = G(t)V(t) : G(t) U (t)H 1 U 0(t) ≡ 0 which is a version of (2), follows directly from (3), and can be formulated 4 Motion of the reduced density operator

t V(t) = V(0) + 1 G(s)V(s) ds i! +0 and by iteration gives

t i V(t) = ←℘ exp G(s) ds V(0) (4) ! − 0 · $ + ' where ←℘ is the chronological ordering operator. Typically (as in QED) H 1 is time-dependent (abruptly/slowly switched on then off): in such cases one i cannot write U(t) = exp (H 0 + H 1)t so (3) fails and one has no alternative ! but to work from (4). − ! " From (2)—which describes the motion of Ψ t in the interaction picture— we get | % i! ∂t ρ(t) = [G(t), ρ(t)] (5.1)

–1 where ρ(t) = Ψ t t Ψ = U 0 (t)ρt U 0(t) is the interaction picture version of the density operator.| % Equiv$ | alently,

t ρ(t) = ρ(0) + 1 [G(s), ρ(s)] ds (5.2) i! +0 Returning with these ideas to the of open systems, it becomes natural to identify H + H with H ; • s e 0 identify H with H . • i 1 Working first in the Schr¨odinger picture, we have

ρ = U(t)ρ U –1(t) i! ∂ ρ = [H, ρ ] t 0 ⇐⇒ t t t where ρt and H both refer to the total (composite) system. Tracing out the environmental terms, we get1

ρ = tr U(t)ρ U –1(t) i! ∂ ρ = tr [H, ρ ] st e 0 ⇐⇒ st t e t

I know from low-dimensional! numerical" experiments that the! spectrum" of ρρst •• is time-dependent, so the transformation ρs0 ρst cannot be unitary (even though the experiments indicate that it is−→-preserving). Breuer & Petruccione (their page 114) draw

ρ ρ U(t , t ) ρ U –1(t , t ) t1 ←−−−−−−−−−−−−−−−−−−→ t2 ≡ 2 1 t1 2 1 ↓ ↓ ρ tr ρ ρ tr ρ st1 ≡ e t1 −−−−−−−−−−−−→ st2 ≡ e t2

1 Compare Heinz-Peter Breuer & Francesco Petruccione, The Theory of Open Quantum Systems (), page 112. Quantum dynamical map 5 to emphasize that, while the transformations T(t , t ) : ρ ρ are clearly 2 1 t1 −→ t2 elements of a with an obvious composition law T(t3, t2)T(t2, t1) = T(t3, t1), –1 obvious identity T(t, t) and obvious inversion law T (t2, t1) = T(t1, t2), the T induced transformations s(t2, t1) : ρst1 ρst2 are not elements of a group: they satisfy the same composition law,−→but are non-invertible. To invert T s(t2, t1) one would, as step #1 (see the diagram), have to invert ρt2 ρst2 which—since the step entails loss of information (is essentially projectiv−→e)—is impossible. The transformations Ts(t2, t1) are elements not of a group but of a semi group. In realistic situations we do not know nearly enough about the quantum of the environment—the generated by H e+H i to permit us to proceed by direct application of the process diagrammed above to T useful information about the process of interest: s(t2, t1) : ρst1 ρst2 . What we need is an approximation scheme that permits us to describ−e→the evolution of ρs in terms of operators that act on Hs, and into which the physics generated by H e and H i enters only in a simplified way, as an “influence.” Equations that accomplish that objective are called “master equations,” and come in several flavors. My own discussion of this subject borrows from the following sources: Maximillian Schlosshauer, Decoherence & the Quantum-to-Classical Transition (), especially Chapter 4 (“Master-equation formulations of decoherence”); H.-P. Breuer &F. Petruccione, The Theory of Open Quantum Systems (), especially 3.2.1–4; U. Weiss, Quantum Dissipative Systems (3rd edition ), 2.3. § § Quantum dynamical map. We work from

–1 ρst = tre U(t)ρ0 U (t) (6) ! " and from the assumption that initially ρ0 factors

ρ = ρ ρ 0 s0 ⊗ e0 We assume moreover—quite unrealistically (!), but in service of our formal objective—that we possess the spectral resolution of ρe0:

ρ = λ φ φ e0 α| α%$ α| α # where the λα are positive real numbers that sum to unity. Let ψβ comprise and orthonormal basis in H , and with its aid construct | % e ! " –1 ρst =tre U(t)ρ0 U (t)

–1 ! = I s "ψβ U(t) ρs0 λα φα φα U (t) I s ψβ ⊗ $ | ⊗ | %$ | ⊗ | % β α # ) * ) # * ) * Use

ρ λ φ φ = I λ φ ρ 1 I λ φ s0 ⊗ α| α%$ α| s ⊗ α| α% s0 ⊗ s ⊗ α$ α| α α ) # * # ) , * *) , * 6 Motion of the reduced density operator to obtain (since ρ 1 = ρ ) s0 ⊗ s0 ) * + ρst = W αβ(t) ρs0 W αβ(t) (7.1) #α,β with λα I s ψβ U(t) I s φα W αβ(t) ⊗ $ | ⊗ | % ≡ (7.2) + + ,λα )I s φα *U (t))I s ψβ * = W (t) ⊗ $ | ⊗ | % αβ The dimensions of,the matrices) that* enter) into the preceding* definitions conform to the following pattern =

where the short sides have length s, the long sides have length s + e. It now follows from ψ ψ = I , normality φ φ = 1 and λ = 1 that β | β%$ β| e $ α| α% α α +- - W αβ(t)W αβ(t) #α,β = λ I φ U +(t) I ψ ψ U(t) I φ α s ⊗ $ α| s ⊗ | β%$ β| s ⊗ | α% α β # ) * ) # * ) * = λ I φ U +(t) I I U(t) I φ α s ⊗ $ α| s ⊗ e s ⊗ | α% α # ) * ) * ) * = λ I φ I I φ α s ⊗ $ α| s+e s ⊗ | α% α # ) * ) * = λ I φ I φ by I I = I and I φ = φ α s ⊗ $ α| s ⊗ | α% s· s s e| α% | α% α # ) *) * = λ I 1 α s ⊗ α # ) * = I s (8) Returning with this information to (7.1), we have tr ρ = tr ρ W + (t)W (t) = tr ρ = 1 st s0 · αβ αβ s0 α,β ! " ! # " ! " Note that we could but need not identify ψα with φα , as Breuer & Petruccione elected to do. | % | % ! " ! " Equations (7) do provide a description of ρ ρ , but presume that s0 +−→ st we possess information—the spectral representation of ρe0, an evaluation of U(t) = exp (i/!)(h s h e + H i) t —that in realistic cases we cannot expect to have. It −is, in this ⊗respect, gratifying to observe that in the absence of ! " system-environmental interaction we have U(t) = U (t) U (t) and (7) reads s ⊗ e Quantum dynamical map 7

ρ = U (t) ρ U+(t) ψ U (t) φ λ φ U+(t) ψ st s s0 s ⊗ $ β| e | α% α$ α| e | β% α,β ) * # = U (t) ρ U+(t) tr U (t) ρ U+(t) s s0 s ⊗ e e0 e + = )U s(t) ρs0 U s (t)* ! "

The motion of ρst has in the absence of interaction become unitary. It is the presence of the α,β that destroys the unitarity of (7). Breuer & P-etruccione write

ρ = W (t) ρ W + (t) V(t)ρ (9) st αβ s0 αβ ≡ s0 #α,β where V(t) is an operator (what Breuer & Petruccione call a “super-operator”) that achieves a certain linear reorganization of the elements of ρs0. Suppose, for example, that Hs is 2-dimensional, and that ρs0 can be represented

ρ0,11 ρ0,12 ρρs0 = • ρ0,21 ρ0,22 • . / Then the upshot of (9) can be described

ρt,11 V11,11(t) V11,12(t) V11,21(t) V11,22(t) ρ0,11 ρ V (t) V (t) V (t) V (t) ρ t,12 = 12,11 12,12 12,21 12,22 0,12  ρt,21   V21,11(t) V21,12(t) V21,21(t) V21,22(t)   ρ0,21   ρt,22   V22,11(t) V22,12(t) V22,21(t) V22,22(t)   ρ0,22        and abbreviated

ρ,ρst = V(t) ρ,ρs0 •• •• Proceeding on the assumption that it is possible to write V(t) = exp L t we are led to a differential equation of “Markovian” form ! " d ρ,ρst = L ρ,ρst dt •• •• This equation illustrates the explicit meaning of the equation that Breuer & Petruccione write d dt ρst = Lρst (10) and call the “Markovian quantum .” We undertake now to develop the explicit meaning of the Lρst. 2 Suppose Hs to be n-dimensional. Let operators Fi, i = 1, 2, . . . , n span the space of linear operators on Hs. Assume without loss of generality that the Fi are orthonormal in the tracewise sense

(F , F ) 1 tr F + F = δ i j ≡ n i j ij ! " 8 Motion of the reduced density operator

Assume more particularly that F1 = I. The remaining basis operators are then necessarily traceless: (F1, Fi=1) = 0 tr Fi=1 . For any linear operator A on # ∼ # Hs we have n2 ! "

A = Fi (Fi, A) i=1 # In particular, we have n2

W αβ = Fi (Fi, W αβ) i=1 # which by (9) gives

n2 + V(t)ρs0 = W αβ(t) ρs0 W αβ(t) i,j=1 # #α,β n2 + = (Fi, W αβ)(Fj, W αβ)∗ Fi ρs0 Fj i,j=1 # #α,β

cij(t) 6 78 9 Taking now into account the unique simplicity of F1 = I, we have

n2 n2 + + Lρst = c˙11ρs0 + c˙i1 Fi ρs0 + c˙1i ρs0 Fj + c˙ij Fi ρs0 Fj i=2 i,j=2 # : ; # All the computational difficulty resides now in the functions c˙ij(t). At t = 0 those become constants a c˙ (0) and the preceding equation reads ij ≡ ij n2 n2 + + Lρs0 = a11ρs0 + ai1 Fi ρs0 + a1i ρs0 Fj + aij Fi ρs0 Fj i=2 i,j=2 # : ; # n2 + + = a11ρs0 + Fρs0 + ρs0 F + aij Fi ρs0 Fj (11) i,j=2 % & # which serves to define the action of the “super-generator” L. Here

n2 n2 F a F = F + = a F+ by the hermiticity of c (t) ≡ i1 i ⇒ 1i i - ij - i=2 i=2 # # Resolving F into its hermitian and anti-hermitin parts F = 1 F + F + + 1 F F + g iH 2 2 − ≡ − we have % & % & Fρ + ρ F + = i Hρ ρ H + gρ + ρ g s0 s0 − s0 − s0 s0 s0 and find that% (11) can be written& % & % & Quantum dynamical map 9

n2 Lρ = i[H, ρ ] + G, ρ + a F ρ F + (12) s0 − s0 { s0} ij i s0 j i,j=2 # with 1 G = 2 a11 I + g From ρ = eLtρ = ρ + t Lρ + st s0 s0 · s0 · · · and the previously established fact that tr ρst = tr ρs0 we conclude from (12) that ! " ! " n2 + 0 = tr 2G + aij Fj Fi ρs0 : all ρs0 i,j=2 $: # ; ' whence n2 G = 1 a F + F − 2 ij j i i,j=2 # Equation (12) now presents what Breuer & Petruccione call the “first standard form” n2 Lρ = i[H, ρ ] + a F ρ F + 1 F + F , ρ (13) s − s ij i s j − 2 { j i s} i,j=2 # : ; of the description of the action achieved by the super-generator L. Further progress requires that we sharpen what we know about the coefficients c (t) (F , W )(F , W )∗ ij ≡ i αβ j αβ #α,β It is, as previously remarked, immediate that c (t) is hermitian. Moreover - ij -

2 vi∗cijvj = (vk Fk, W αβ) ! 0 : all complex vectors v i,j α,β <: k ;< # # < # < < < so the eigenvalues of c (t) must all be non-negative. The same can, of course, - ij - be said of aij . And of the sub-matrix that results from restricting the range of i and j,-as i-s done in (13). Let u be elements of the that diagonalizes a : ip - ij-

λ2 0 . . . 0 0 λ3 . . . 0 aij = uipΛpqu¯jq with Λpq =  . . . .  - - . . . . p,q . . . . #    0 0 . . . λn2    10 Motion of the reduced density operator

Equation (13) becomes

n2 Lρ = i[H, ρ ] + u Λ u¯ F ρ F + 1 F + F , ρ s − s ip pq jq i s j − 2 { j i s} i,j,p,q=2 # : ; n2 = i[H, ρ ] + λ δ u F ρ u¯ F + 1 u¯ F +u F , ρ − s p pq ip i s jq j − 2 { jq j ip i s} i,j,p,q=2 # : ; n2 = i[H, ρ ] + λ A ρ A+ 1 A+ A ρ 1 ρ A+ A (14) − s p p s p − 2 p p s − 2 s p p p=2 # : ; with n2

A p = uip Fi i=2 # 2 The operators A p, p = 2, 3, . . . , n are called “Lindblad” operators, and

n2 d ρ = i[H, ρ ] + λ A ρ A+ 1 A+ A ρ 1 ρ A+ A (15) dt st − st p p st p − 2 p p st − 2 st p p p=2 # : ;

D(ρst) is called the “Lindblad equation.”6 2 The H term78generates a unitary motion9 which is distinct from that which in the absence of system-environmental interaction would have been generatd by H s; the interaction term H i was seen 1 + above to enter into the construction of H = i 2 F F . It is the “dissipator” D(ρ ) that accounts for the non-unitarity of ρ − ρ . st %s0 +−→ &st Much freedom attended our selection of an orthonormal basis

I, F 2, F 3, . . . F n2 in the “Liouville space” of linear! operators on Hs", so the expression on the right side of (15) is highly non-unique. If, for example, we write

λp A p = upq µq B q q , # , we obtain n2 D(ρ ) = λ A ρ A+ 1 A+ A ρ 1 ρ A+ A (16) st p p st p − 2 p p st − 2 st p p p=2 # : ; n2 + = upq µq B qρstu¯pr µr B r p,q,r=2 # : , , 1 u¯ µ B+ u µ B ρ 1 ρ u¯ µ B+ u µ B − 2 pr r r pq q q st − 2 st pr r r pq q q , , , , ; 2 Breuer & Petruccione cite G. Lindblad, “On the generator of quantum mechanical semigroups,” Commun. Math. Physics 48, 119–130 (1976). Quantum dynamical map 11 which, if we impose the unitarity assumption p upqu¯pr = δqr, becomes - n2 D(ρ ) = µ B ρ B+ 1 B+ B ρ 1 ρ B+ B st q q st q − 2 q q st − 2 st q q q=2 # : ; which is structurally identical to (16). Or consider

A A + a I p +−→ p p which (working again from (16)) sends

n2 D(ρ ) D(ρ ) + λ a ρ A+ 1 A+a ρ 1 ρ A+a st +−→ st p p st p − 2 p p st − 2 st p p p=2 # $ : ; + A ρ a¯ 1 a¯ A ρ 1 ρ a¯ A + 0 p st p − 2 p p st − 2 st p p : ;' ↓ n2 D(ρ ) + 1 λ a [ρ , A+] a¯ [ρ , A ] st 2 p p st p − p st p p=2 # $ ' ↓ n2 D(ρ ) + ρ , 1 λ a A+ a¯ A st st 2 p p p − p p p=2 ) # : ;* The additive term can be absorbed into a redefinition of the effective Hamiltonian:

n2 H H + i 1 λ a A+ a¯ A (17) +−→ 2 p p p − p p p=2 # : ; Note the manifest hermiticity of the added term. Weiss3 provides this simpler-looking version of the Lindblad equation (15):

n2 d ρ = i [H, ρ ] + 1 λ [A ρ , A+] + [A , ρ A+] dt st − st 2 p p st p p st p p=2 # : ;

D(ρst) 6 78 9

3 Ulrich Weiss, Quantum Dissipative Systems (3rd edition ), page 11. 12 Motion of the reduced density operator

notational remark: Breuer & Petruccione’s notation D(ρs) serves well enough to signify a “ of (the matrix elements of) an operator, though D(ρs) would better emphasize that we are talking about an operator valued function of an operator. One would expect in that same spirit to write V(ρs, t) and L(ρs) where Breuer & Petruccione elect to write V(t)ρs and Lρs, even though the “super-operators” V(t) and L are defined always by their functional action, never as stand-alone objects. It’s my guess that they do so to motivate the train of thought that follows from writing V(t) = exp V t . In these respects the matrix notation to which I alluded on page 7 provides a more frankly informative account of ! " the situation.

Alternative derivation of the Lindblad equation. The results obtained in the preceding section are formally exact, but acquire their exactitude from the seldom/never justified assumption that we possess an exact description of

U(t) = exp (i/!)[H s + H e + H i]t − ! " In their 3.3.1 Breuer & Petruccione develop a line of argument that avoids that strong assumption—an§ argument that owes its success to certain simplifying assumptions (which is why Weiss3 considers the Lindblad equation to be valid “in the Born-Markov approximation). We work in the interaction picture, where operators generally (and H i in particular) move as directed by [H s + H e] and where the of the composite system moves as directed by the (now time-dependent) interaction Hamiltonian4

H (t) U +(t)H U (t) with U (t) exp i [H + H ]t i ≡ 0 i 0 0 ≡ − s e ) * Thus d ρ(t) = i[H (t), ρ(t)] dt − i ↓ t ρ(t) = ρ(0) i [H (s), ρ(s)] d − i +0 Insert the latter into the former (inspired idea!)

t d ρ(t) = i[H (t), ρ(0)] [H (t), [H (s), ρ(s)]] ds dt − i − i i +0 trace out the environmental degrees of freedom, assume (on what grounds?) that tre[H i(t), ρ(0)] = 0 s

4 Here I revert to my former asumption that we have adopted units in which ! = 1. Alternative derivation of the Lindblad equation 13 to obtain

t d ρ (t) = tr [H (t), [H (s), ρ(s)]] ds dt s − e i i +0

↓ t weak coupling, or tre[H i(t), [H i(s), ρs(s) ρe]] ds ≈ − 0 ⊗ born approximation + $ ↓ t time-localized tr [H (t), [H (s), ρ (t) ρ ]] ds ≈ − e i i s ⊗ e “Redfield equation” +0 ( To achieve the markoff approximation we must eliminate all reference to history (here: the “start time,” which in the previous discussion was built into the structure of U(t, 0)), To that end, write s = t u −

↓ 0 = + tre[H i(t), [H i(t u), ρs(t) ρe]] du t − ⊗ + t = tr [H (t), [H (t u), ρ (t) ρ ]] du − e i i − s ⊗ e +0 and let the upper limit : ↑ ∞

∞ = tr [H (t), [H (t u), ρ (t) ρ ]] du − e i i − s ⊗ e +0 This Born-Markoff master equation approximates the Lindblad equation (15) in a sense (i.e., under physical conditions) that Breuer & Petruccione attempt to summarize on their page 131, but which I am not yet in position to discuss.5

5 For a helpful discussion of this entire topic, see the class notes of Andrew Fisher at http://www.cmmp.ucl.ac.uk/ajf course notes/node35.html. Fisher cites Breuer & Petruccione’s Chapter 3 (esp∼ ecially 3.2) and also lecture notes by John Preskill ( class at Caltec§ h).