Eur. Phys. J. B 21, 553–560 (2001)
THE EUROPEAN PHYSICAL JOURNAL B EDP Sciences c Societ`
a Italiana di Fisica Springer-Verlag 2001
Master and Langevin equations for electromagnetic dissipation and decoherence of density matrices Z. Habaa and H. Kleinertb Institut f¨ ur Theoretische Physik, Freie Universit¨ at Berlin, Arnimallee 14, 14195 Berlin, Germany Received 29 November 2000 and Received in final form 11 February 2001 Abstract. We set up a forward – backward path integral for a point particle in a bath of photons to derive a master equation for the density matrix which describes electromagnetic dissipation and decoherence. We also derive the associated Langevin equation. As an application, we recalculate the Wigner-Weisskopf formula for the natural line width of an atomic state at zero temperature and find, in addition, the temperature broadening caused by the decoherence term. Our master equation also yields the correct Lamb shift of atomic levels. The two equations may have applications to dilute interstellar gases or to few-particle systems in cavities. PACS. 03.65.-w Quantum mechanics – 03.65.Yz Decoherence; open systems; quantum statistical methods – 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.45.-a Nonlinear dynamics and nonlinear dynamical systems
1 Introduction The time evolution of a quantum-mechanical density matrix ρ(x+a , x−a ; ta ) of a particle coupled to an external electromagnetic vector potential A(x, t) is determined by a forward – backward path integral [1] (x+b , tb |x+a , ta )(x−b , tb |x−a , ta )∗ ≡ U (x+b , x−b , tb |x+a , x−a , ta ) ( Z " Z i tb M 2 = Dx+ Dx− exp x˙ + − x˙ 2− − V (x+ ) ~ ta 2 #) e e +V (x− ) − x˙ + A(x+ , t) + x˙ − A(x− , t) , (1.1) c c where x+ (t) and x− (t) are two fluctuating paths connecting the initial and final points x+a and x+b , and x−a and x−b , respectively. In terms of this expression, the density matrix ρ(x+b , x−b ; tb ) at a time tb is found from that at an earlier time ta by the integral Z ρ(x+b , x−b ; tb ) =
dx+a dx−a
× U (x+b , x−b , tb |x+a , x−a , ta )ρ(x+a , x−a ; ta ). (1.2) a On leave from Institute of Theoretical Physics, University of Wroclaw, Poland b e-mail:
[email protected] URL: http://www.physik.fu-berlin.de/~kleinert
The vector potential RA(x, t) appearing in the electromagnetic action Aem = d4 x(E2 − B2 )/2c in the radiation ˙ gauge via E = A/c, B ≡ ∇ × A, is a superposition of oscillators Xk (t) of frequency Ωk = c|k| in a volume V : A(x, t) = X
X k
≡V
k
Z
fk (x)Xk (t),
eikx , fk (x) = p 2V Ωk /c
d3 k · (2π)3
(1.3)
These oscillators are assumed to be in equilibrium at a finite temperature T , where we shall write 0 their time-ordered correlation functions as Gij kk0 (t, t ) = 0 ˆ i (t), X ˆ j 0 (t0 )i = δ ij tr ≡ δkk0 (δ ij − hTˆX k −k kk0 GΩk (t, t ) i j 2 0 k k /k )GΩk (t, t ), the transverse Kronecker symbol reP sulting from the sum h=± i (k, h)j∗ (k, h) over the two polarization vectors of the vector potential A(x, t). For a single oscillator of frequency Ω, one has for t > t0 : 1 [AΩ (t, t0 ) + CΩ (t, t0 )] 2 Ω 0 ~ cosh 2 [~β − i(t − t )] = , ~Ωβ 2M Ω sinh 2
GΩ (t, t0 ) =
t > t0 (1.4)
which is the analytic continuation of the periodic imaginary-time Green function to τ = it. The decomposition into AΩ (t, t0 ) and CΩ (t, t0 ) distinguishes real and imaginary parts, which are commutator and
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anticommutator functions of the oscillator at temperˆ ˆ 0 )]iT and AΩ (t, t0 ) ≡ ature T : CΩ (t, t0 ) ≡ h[X(t), X(t 0 ˆ ˆ )]iT , respectively. The thermal average of the h[X(t), X(t evolution kernel (1.1) is then given by the forward – backward path integral Z Z U (x+b , x−b , tb |x+a , x−a , ta ) = Dx+ (t) Dx− (t) Z tb M 2 i dt (x˙ + − x˙ 2− ) − (V (x+ ) − V (x− )) × exp ~ ta 2 i + AFV [x+ , x− ] . (1.5) ~ FV
where exp{iA [x+ , x− ]/~} is the Feynman-Vernon influence functional . The influence action AFV [x+ , x− ] is the sum of a dissipative and a fluctuating part AFV D [x+ , x− ] and AFV [x , x ], respectively, whose explicit forms are + − F Z Z ie2 [x , x ] = AFV dt dt0 Θ(t − t0 ) + − D 2~c2 h × x˙ + Cb (x+ t, x0+ t0 )x˙ 0+ − x˙ + Cb (x+ t, x0− t0 )x˙ 0− i − x˙ − Cb (x− t, x0+ t0 )x˙ 0+ − x˙ − Cb (x− t, x0− t0 )x˙ 0− (1.6) and AFV F [x+ , x− ]
ie2 = 2~c2
Z
Z dt
where x± , x0± are short for x± (t), x± (t0 ), and Cb (x− t, x0− t0 ), Ab (x− t, x0− t0 ) are 3 × 3 commutator and anticommutator functions of the bath of photons. They are sums of correlation functions over the bath of the oscillators of frequency Ωk |, each contributing with a weight 0 fk (x)f−k (x0 ) = eik(x−xR) c/2ΩkV (the normalization following from the action d4 x(E2 − B2 )/2c). Thus we may write D E X ˆ i (t), X ˆ j (t0 )] Cbij (x t, x0 t0 ) = f−k (x)fk (x0 ) [X −k k 0 3
dω d k ij tr ik(x−x0) σk (ω 0 )δkk e sin ω 0 (t − t0 ), 4 (2π) (1.8)
= −ic2 ~
0 0 Aij b (x t, x t ) =
T
k
X k
f−k (x)fk (x0 )
Dn
1 [A(x, x0 ) + C(x, x0 )] 2 Z 0 0 ic2 ~ dω d3 k = e−i[ω(t−t )−k(x−x )] 2 4 2 (2π) ω − Ωk + iη Z d4 k ik(x−x0 ) i~ =c e · (1.11) (2π)4 k 2 + iη
G(x, x0 ) =
where η is an infinitesimally small number > 0. We shall now focus attention upon systems which are so small that the effects of retardation can be neglected. Then we can ignore the x-dependence in (1.8) and (1.9) and find Cbij (x t, x0 t0 ) ≈ Cbij (t, t0 ) = i
oE ˆ i (t), X ˆ j (t0 ) X −k k
T
Z dω 0 d3 k ~ω 0 ik(x−x0) 0 ijtr = c2 ~ σ (ω )δ coth e cos ω 0 (t−t0 ), k kk (2π)4 2kB T (1.9) where σk (ω 0 ) is the spectral density contributed by the oscillator of momentum k: 2π σk (ω 0 ) ≡ [δ(ω 0 − Ωk ) − δ(ω 0 + Ωk )]. (1.10) 2Ωk
~ 2 ij δ ∂t δ(t − t0 ). (1.12) 2πc 3
Inserting this into (1.6) and integrating by parts, we obtain two contributions. The first is a diverging term Z ∆M tb ∆Aloc [x+ , x− ] = dt (x˙ 2+ − x˙ 2− )(t), (1.13) 2 ta where
dt0 Θ(t − t0 )
h × x˙ + Ab (x+ t, x0+ t0 )x˙ 0+ − x˙ + Ab (x+ t, x0− t0 )x˙ 0− i − x˙ − Ab (x− t, x0+ t0 )x˙ 0+ + x˙ − Ab (x− t, x0− t0 )x˙ 0− , (1.7)
Z
At zero temperature, we recognize in (1.8) and (1.9) twice the imaginary and real parts of the Feynman propagator of a massless particle for t > t0 , which in four-vector notation with k = (ω/c, k) and x = (ct, x) reads
e2 ∆M ≡ − 2 c
Z
dω 0 d3 k σk (ω 0 ) ij tr e2 δ = − kk (2π)4 ω0 3π 2 c3
Z
∞
dk. 0
(1.14) diverges linearly. This simply renormalizes the kinetic terms in the path integral (1.5), renormalizing them to Z i tb Mren 2 dt x˙ + − x˙ 2− . (1.15) ~ ta 2 By identifying M with Mren this renormalization may be ignored. The second term has the form Z M tb ¨ − )R (t), AFV [x , x ] = −γ dt (x˙ + − x˙ − )(t)(¨ x+ + x + − D 2 ta (1.16) with the friction constant γ≡
e2 2 α = , 3 6πc M 3 ωM
(1.17)
where α ≡ e2 /~c ≈ 1/137 is the fine-structure constant and ωM ≡ M c2 /~ the Compton frequency associated with the mass M . In contrast to the ordinary friction constant, this has the dimension 1/frequency. Note that the retardation enforced by the Heaviside function in the exponent of (1.6) removes the left-hand half of the δ-function. It expresses the causality of the dissipation forces, which is crucial for producing a probability conserving time evolution of the probability distribution [2]. The superscript R in (1.16) accounts for this by
Z. Haba and H. Kleinert: Master equation for electromagnetic dissipation
¨ − )(t) is slightly indicating that the acceleration (¨ x+ + x shifted with respect to the velocity factor (x˙ + − x˙ − )(t) towards an earlier time. We now turn to the anticommutator function. Inserting (1.10) and the friction constant γ from (1.17), it becomes e2 Ab (x t, x0 t0 ) ≈ 2γkBT K(t, t0 ), c2 where 0
Z
0
K(t, t ) = K(t − t ) ≡
∞
−∞
(1.18)
0 0 dω 0 K(ω 0 )e−iω (t−t ) , (1.19) 2π
with a Fourier transform K(ω 0 ) ≡
~ω 0 ~ω 0 coth , 2kB T 2kB T
The function K(ω 0 ) has the normalization K(0) = 1, giving K(t − t0 ) a unit time integral: Z ∞ dt K(t − t0 ) = 1. (1.22) −∞
Thus K(t − t0 ) may be viewed as a δ-function broadened by quantum fluctuations. With the function K(t, t0 ), the fluctuation part of the influence functional in (1.5–1.7) becomes
Z ×
tb
w =i 2~
Z
tb
dt ta
dt0 (x˙ + − x˙ − )(t) K(t, t0 ) (x˙ + − x˙ − )(t0 ). (1.23)
ta 0
Here we have used the symmetry of the function K(t, t ) to remove the Heaviside function Θ(t−t0 ) from the integrand, extending the range of t0 -integration to the entire interval (ta , tb ). We also have introduced the constant w ≡ 2M kB T γ,
where the last term becomes local for high temperatures, since K(t, t0 ) → δ(t − t0 ). This is the closed-time path integral of a particle in contact with a thermal reservoir. For moderately high temperature, we should include also the first correction term in (1.21) which adds to the exponent an additional term Z tb w ¨ − )2 . − dt(¨ x+ − x (1.26) 24(kBT )2 ta In the classical limit, the last term in (1.25) squeezes the forward and backward paths together. The density matrix (1.25) becomes diagonal, and the γ-term describes classical radiation damping.
(1.20)
whose high-temperature expansion starts out like 2 1 ~ω 0 . (1.21) K(ω 0 ) ≈ K HT (ω 0 ) ≡ 1 + 3 2kB T
AFV F [x+ , x− ]
555
(1.24)
for brevity. At very high temperatures, the time evolution amplitude for the density matrix is given by the path integral Z Z U (x+b , x−b , tb |x+a , x−a , ta ) = Dx+ (t) Dx− (t) Z tb i M 2 2 × exp dt (x˙ − x˙ − )−(V (x+ )−V (x− )) ~ ta 2 + Z tb i ¨ − )R × exp − M γ dt (x˙ + − x˙ − )(¨ x+ + x 2~ ta Z tb w − 2 dt (x˙ + − x˙ − )2 , (1.25) 2~ ta
2 Master equation for time evolution of density matrix We now derive a Schr¨odinger-like differential equation describing the evolution of the density matrix ρ(x+a , x−a ; ta ) in equation (1.2). In the standard derivation of such an equation [3] one first makes the last term local via a quadratic completion involving a fluctuating noise variable η(t). Then one goes over to a canonical formulation of the path integral (1.25), by rewriting it as a path integral Uη (x+b , x−b , tb |x+a , x−a , ta ) = Z Z Z Z Dp− Dp+ Dx+ (t) Dx− (t) 3 (2π) (2π)3 Z tb i × exp dt [p+ x˙ + −p− x˙ − −Hη (p+ , p− , x+ , x− )] . ~ ta (2.1) Then Uη (x+b , x−b , tb |x+a , x−a , ta ) satisfies the differential equation ˆ η Uη (x , y , t|xa , ya , ta ). i~∂t Uη (x , y , t|xa , ya , ta ) = H (2.2) The same equation is obeyed by the density matrix ρ(x+ , x− ; ta ). At high temperatures where the action in the path integral (1.25) is local, we can set up directly a Hamiltonian without the noise-averaging procedure. However, the standard procedure of going to a canonical formulation is not applicable because of the high time derivatives of x(t) in the action of (1.25). They can be transformed into canonical momentum variables only by introducing several aux˙ b≡x ¨ , . . . [4,5]. For iliary independent variables v ≡ x, small dissipation, which we shall consider, it is preferable to proceed in another way by going first to a canonical formulation of the quantum system without electromagnetism, and include the effect of the latter recursively. For simplicity, we shall treat only the local limiting form of the last term in (1.25). In this limit, we define a Hamilton-like
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operator as follows: ˆ≡ 1 p ˆ 2+ − p ˆ 2− + V (x+ ) − V (x− ) H 2M Mγ ˆ ˆ˙ + − x ˆ˙ − )(x ˆ¨ + + x ˆ¨ − )R − i w (x ˆ˙ − )2 . (x˙ + − x + 2 2~ (2.3) ˆ˙ x ˆ¨ are abbreviations for the commutators Here x, ˆ x ˆ˙ ≡ i [H, ˆ ], x ~
ˆ¨ ≡ i [H, ˆ x]. ˆ˙ x ~
(2.4)
A direct differentiation of equation (1.25) over time leads to the conclusion that the density matrix ρ(x+ , x− ; ta ) satisfies the time evolution equation ˆ i~∂t ρ(x+ , x− ; ta ) = Hρ(x + , x− ; ta ).
(2.5)
At moderately high temperatures, we also include a term coming from (1.26) H1 ≡ i
w~ ˆ¨ + − x ˆ¨ − )2 . (x 24(kB T )2
(2.6)
For systems with friction caused by a conventional heat bath of harmonic oscillators as discussed by Caldeira and Leggett [6], the analogous extra term was shown by Diosi [7] to bring the Master equation to the general Lindblad form [8] which ensures positivity of the probabilities resulting from the solutions of (2.5). It is useful to re-express (2.5) in the standard quantummechanical operator form where Pthe density matrix has a bra–ket representation ρˆ(t) = mn ρnm (t)|mihn|. Let us denote the initial Hamilton operator of the system in (1.1) ˆ =p ˆ 2 /2M + Vˆ , then equation (2.5) with the term by H (2.6) takes the operator form ˆ ρˆ ≡ [H, ˆ ρˆ] + M γ x ˆ˙ x ˆ¨ ρˆ − ρˆx ˆ¨ x ˆ˙ + x ˆ˙ ρˆ x ˆ¨ − x ˆ¨ ρˆ x ˆ˙ i~∂t ρˆ = H 2 iw ˆ ˆ iw~ ¨ˆ , [x ¨ˆ , ρˆ]]. ˙ [x, ˙ ρˆ]] − − [x, [x (2.7) 2~ 24(kB T )2 The operator order in the terms in parentheses need ex¨ ± with respect planation. It is fixed by the retardation of x to x˙ ± in (2.3), which implies that the associated operator ˆ¨ (t) has a time argument which lies slightly before that x ˆ˙ ± , thus acting upon ρˆ before the operator of velocity of x ˆ˙ This puts it to the right of x, ˆ˙ i.e., next to ρˆ. On the x. right-hand side of ρˆ, the time runs in the opposite direcˆ¨ must lie left of x, ˆ˙ again next to ρˆ. In tion such that x this way we obtain an operator order which ensures that equation (2.7) conserves the total probability. This property and the positivity of ρˆ are guaranteed by the observation, that equation (2.7) can be written in the extended Lindblad form [8] 2 X i ˆ 1 ˆ ˆ† 1 ˆ ˆ† † ˆ ˆ ∂t ρˆ = − [H, ρˆ] − Ln Ln ρˆ + ρˆLn Ln − Ln ρˆLn . ~ 2 2 n=1 (2.8)
with the two Lindblad operators √ √ wˆ 3w ˆ ~ ˆ ˆ ˆ ˙ ¨ . (2.9) L1 ≡ x, L2 ≡ x˙ − i x 2~ 2~ 3kB T ˆ˙ x ˆ¨ ρˆ Note that the operator order prevents the term x from being a pure divergence. If we rewrite it as a sum of ˆ˙ x ˆ¨ ]/2+{x, ˆ˙ x ˆ¨ }/2, a commutator and an anticommutator, [x, then the latter term is a pure divergence, and we can think of the first two γ-terms in (2.7) as being due to an addiˆ tional anti-Hermitian term in the Hamilton operator H, the dissipation operator ˆ γ = γM [x, ˆ˙ x ˆ¨ ]. H 4
(2.10)
ˆ p ˆ ] = 0, one For a free particle with V (x) ≡ 0 and [H, ˆ˙ ± = p ˆ ± /M to all orders in γ, such that the time has x evolution equation (2.7) becomes ˆ ρˆ] − i~∂t ρˆ = [H,
iw [ˆ p, [ˆ p, ρˆ]]. 2M 2 ~
(2.11)
In the P momentum representation of the density matrix ρˆ = pp0 ρpp0 |pihp0 |, the last term simplifies to −iΓ ≡ −iw(p − p0 )2 /2M 2 ~2 multiplying ρˆ, which shows that a free particle does not dissipate energy by radiation, and that the off-diagonal matrix elements decay with the rate Γ . In general, equation (2.3) is an implicit equation for ˆ For small e2 it can be solved the Hamilton operator H. ˆ˙ by approximately in a single iteration step, replacing x ˆ ˆ ¨ p/M and x = −∇V /M in equation (2.7). The validity of this iterative procedure is most easily proven in the time-sliced path integral. The final slice of infinitesimal width reads Z dp+ (tb ) U (x+b , x−b , tb |x+a , x−a , tb − ) = (2π)3 Z dp− (tb ) i{p+ (tb )[x+ (tb )−x+ (tb −)]−p− x˙ − −H(tb )}/~ × e . (2π)3 (2.12) Consider now a term of the generic form F˙ + (x+ )F− (x− ) in H. When differentiating U (x+b , x−b , tb |x+a , x−a , tb − ) with respect to the final time tb , the integrand receives a factor −H(tb ). The term F˙+ (x+ )F− (x− ) in H has the explicit form −1 [F+ (x+ (tb )) − F+ (x+ (tb − ))] F− (x− (tb )). It it can be taken out of the integral, yielding −1 [F+ (x+ (tb ))U − U F+ (x+ (tb − ))] F− (x− (tb )). (2.13) ˆ ≈ In operator language, the amplitude U is equal to U ˆ ˙ 1 − iH/~, such the term F+ (x+ )F− (x− ) in H yields an operator i i hˆ ˆ H, F+ (x+ ) F− (x− ) (2.14) ~
Z. Haba and H. Kleinert: Master equation for electromagnetic dissipation
in the differential operator for the time evolution. ¨ we have to For functions of the second derivative x split off the last two time slices and convert the two intermediate integrals over x into operator expressions, which ˆ with x ˆ, obviously leads to the repeated commutator of H and so on.
3 Line width Let us apply the master equation (2.7) to atoms, where V (x) is the Coulomb potential, assuming it to be initially in an eigenstate |ii of H, with a density matrix ρˆ(0) = |iihi|. Since atoms decay rather slowly, we may treat the γ-term in (2.7) perturbatively. It leads to a time derivative of the density matrix γ ˆ p ˆ] p ˆ ρˆ(0)|ii hi|[H, ~M X γ = ωif hi|p|f ihf |p|ii M f 6=i X 3 = −M γ ωif |xf i |2 ,
∂t hi|ˆ ρ(t)|ii = −
(3.1)
f
where ~ωif ≡ Ei − Ef , and xf i ≡ hf |x|ii are the matrix elements of the dipole operator. An extra width, which we have not seen discussed in the literature so far, comes from the last two terms in (2.7): w w hi|p2 |ii − hi|p˙ 2 |ii M 2 ~2 12M 2 (kB T )2 " # 2 X ~2 ωif 2 = −w ωif 1 + |xf i |2 . (3.2) 2 12(k T ) B n
two terms accompanied by the Bose occupation function account for induced emission and absorption. For high and intermediate temperatures, (3.4) has the expansion Z 2kB T 4π dω 0 d3 k π 0 −1 ~ 2δ(ω − Ωk ) + 3 (2π)4 2M Ωk ~Ωk 0 0 1 ~Ωk + [δ(ω 0 − Ωk ) + δ(ω 0 + Ωk )] e−iω (t−t ) . (3.5) 6 kB T The first term in curly brackets is due P to spontaneous 3 |xf i |2 to emission. It contributes a term −2M γ f 0, since the δ-function allows only for decays. Second, there is an extra factor 2. Indeed, by comparing (3.3) with (3.5) we see that the spontaneous emission receives equal contributions from the 1 and the coth(~ω 0 /2kBT ) in the curly brackets of (3.3), i.e., from dissipation and fluctuation terms Cb (t, t0 ) and Ab (t, t0 ). Thus our master equation yields for the natural line width of atomic levels the equation X 3 Γ = 2M γ ωif |xf i |2 , (3.6) f
