Lecture 2: Open quantum systems

Transcription

1 Phys 769 Selected Topics in Condensed Matter Physics Summer 21 Lecture 2: Open quantum systems Lecturer: Anthony J. Leggett TA: Bill Coish 1. No (micro- or macro-) system is ever truly isolated U = S + E (+C) (1) U = universe, S = system, E = environment, C = control. All of these are in principle quantum-mechanical. However: 2. When can we treat C (and possibly part of E) as a deterministic classical object? Crudely speaking, when according to a full quantum-mechanical calculation a (substantial) change in the state of S is not accompanied by any substantial change in C Ex: Creation of polarization of an atom. s ˆΠ s = p ˆΠ p =. s ˆΠ p = p ˆΠ s. (2) ( ) must create a quantum superposition α s +β p, more generally ˆΠ = Tr ˆρˆΠ. However, Ψ in = γ s Ψ f = α γ s + β p (3) ( ) α 2 ˆρ f Tr γ ˆρ U = ˆΠ =! (4) β 2 solution: Ψ in = coh s (5) then Ψ f = α coh s + β (â coh ) p + ( ) (6) coh (α s + β p ) ˆΠ. (7) where is orthogonal to coh. If C is approximately in a coherent state, then a coh N 1/2. (Ex: for typical radio wave, N 1 3 ). 3. When can we treat E in terms of a random classical force? Suppose E is known to be harmonic (e.g. blackbody radiation field) and moreover to be in thermal equilibrium. Then there exists a theorem: 1

2 Any thermal equilibrium state of a simple harmonic oscillator, or set of simple harmonic oscillators, can be represented as an incoherent mixture of coherent states [Digression: P-representation of quantum oscillators]. If k B T ω, then coherent states represented in mixture are mainly α 1 (largeamplitude), so they can be treated classically. Thus, we can specify the effects of E by a classical statistical representation. NB: It is a sufficient set of conditions that (a) E is harmonic, and (b) k B T ω, relevant ω. It is not clear that the first condition is necessary. Ex: classical Brownian motion (Einstein, 195): Mẍ + ηẋ (+Mωx) 2 = F(t) classical fluctuating force (8) Here, ηẋ is a steady force due to E. For consistency (want to ensure Mẋ 2 = k B T in thermal equilibrium) we must set F(t) =, F(t)F(t ) = 2ηk B Tδ(t t ). (9) Another example: spin subject to a fluctuating magnetic field (note in this case S is definitely quantum!). 4. Quantum system in contact with a quantum environment NB: Attempts in literature to describe S by phenomenological equation valid only (if at all) for a simple harmonic oscillator. ( most semiclassical system ). In general, we must treat S and E as interacting, highly entangled, systems. In principle: start with ˆρ U (possibly ˆρ S ˆρ E, though see below), evolve ˆρ U with Û(t) exp iĥt, Ĥ ĤS + ĤE + ĤSE, trace over E to obtain the reduced density matrix ˆρ S. In practice, there are two major problems: (1) do we know ĤSE (or for that matter, Ĥ S )? (2) can we carry out the calculation explicitly? For the moment, assume we do know ĤSE ˆV. 5. The weak-coupling limit: The quantum master equation Situation characteristic of quantum oscillators: may apply to some condensed-matter systems (e.g. nuclear spins coupled to phonons). Return below to how weak coupling ˆV must be 2 (1)

3 The usual choice of initial density matrix of U: ˆρ U = ˆρ (t) E ρ(i) S, ˆρ(t) E thermal density matrix corresponding to ĤE. ˆρ (i) S approximation, the reduced density matrix is ˆρ S () ˆρ (i) S. Consider evaluation of ˆρ U (t): i ˆρ U t ] = [Ĥ, ˆρU = equilibrium is arbitrary. With this factorization (Ĥ ĤS + ĤE + ĤSE); ĤSE ˆV (11) Convenient to transform to interaction representation (ψ exp then i ˆρ U t Iterate this equation twice: ˆρ U (t) = ˆρ U () + 1 i ] = [ˆV (t), ˆρU ˆV (t) e i(ĥe+ĥs)t ˆV e i(ĥe+ĥs)t [ˆV ] (t ), ˆρ U () 1 2 and take the derivative with respect to t: dˆρ U = 1 i [ˆV (t), ˆρU ()] 1 2 [ ] i(ĥs + ĤE)t ψ), (12) [ˆV ]] (t ), [ˆV (t ), ˆρ U (t ) [ˆV ]] (t), [ˆV (t ), ˆρ U (t ) This is still exact, and we can take the trace over E to obtain an equation for ˆρ S. (Note that the time derivative and the trace over ) E commute!). For most practical ) cases, we ) find that Tr E ˆV (t)ˆρu () = Tr E (ˆV (t)ˆρ t vanishes (Tr E E (ˆV (t)ˆρ t E = Tr E (ˆV ˆρ t E since ˆρ t E (t) = ˆρt E ), so: dˆρ S = 1 2Tr E ˆρ (i) S We now make two important approximations: [ˆV ]] (t), [ˆV (t ), ˆρ U (t ) (1) Weak-coupling approximation: we can neglect entanglement between S and E, i.e., put (13) (14) (15) ˆρ U (t ) ˆρ S (t )ˆρ E (t ) ˆρ S (t)ˆρ t E (since bath is large ) (16) dˆρ S = 1 ]] 2 Tr E [ˆV (t), [ˆV (t ), ˆρ t E ˆρ S (t ) (17) Typically, the interaction ˆV (t) will involve a sum of terms each of which is a product of some operator ˆQ i of the system and an operator ˆΩ i of the environment. Because the trace of a commutator vanishes identically, most of the terms 3

4 arising from the double commutator in Eq. (17) are zero, and one ends up very schematically with an equation of the form dˆρ S = 1 2 ij where  ˆB Tr ˆBˆρ t E. Ω i (t)ω j (t ) [ [ ]] ˆQi (t), ˆQi (t ), ˆρ S (t ) (2) Markovian approximation: we suppose that the quantity Ω i (t)ω j (t ) has some (18) characteristic range τ c in (t t ). Then if ˆρ S (t) is assumed to vary slowly over τ c, we can approximate ˆρ S (t ) by ˆρ S (t) and extend the integral over t to. Thus, again schematically, dˆρ S = 1 2 ij [ [ ]] K ij ˆQi, ˆQj, ˆρ S (t) (19) where ˆQ i and ˆQ j are now the Schrödinger-representation (time-independent) operators and K ij Ω i ()Ω j (t ) exp iω ij t [needs amplification] (2) The best-known example in quantum optics is the situation of a single oscillator of frequency Ω coupled to cavity modes: then the double commutator gives rise to a characteristic structure dˆρ S (t) a ˆρ S a 1 2 a a ˆρ S 1 2 ˆρ Sa a + (21) Note that in general no evolution equation of the form of the quantum master equation need exist. 6. What if we do not know the form of ĤSE (or perhaps of ĤE) in detail? Let q, p denote the coordinate and canonical momentum of S, and ξ i, π i those of E Ĥ S ĤS(p, q) (22) Ĥ E ĤE(ξ i, π i ) (23) Ĥ SE ĤSE(p, q; ξ i, π i ) (24) Suppose that for relevant values of p and q, only one degree-of-freedom of the environment is only weakly perturbed. Then, trivially, (cf. 19 th -century oscillator model of the atom) Ĥ E i ( ˆp 2 /2m i + 1 ) 2 m iωi 2 ˆx 2 i (25) 4

5 What about ĤSE? In this limit, quite generally, Ĥ SE = i {F i (p, q)ˆx i + G i (p, q)ˆp i } + Φ(p, q) (26) Φ(p, q) may depend on oscillator parameters, but not on {x i, p i }. (F i, G i are related to the master equations of the original interaction). We can motivate the form of F i (p, q), G i (p, q) (a) from symmetries: e.g. if time-reversal invariant, then (can choose degrees of freedom of x i, p i such that) F i (p, q) is even under time reversal, G i (p, q) odd. (b) from requirements on the classical equations of motion: e.g. suppose the classical damped equation of motion is of the form M q + η q + V (q) = (27) then barring pathology G i (p, q) = and F i (p, q) = qc i. This then leads to a simple form Generalization: Ĥ SE = q i C i x i (+counterterm) (28) (a) η = η(q) but not frequency-dependent, then C i f i (q). (b) η is frequency-dependent, but not η(q), then a simple form is OK (but J(ω) [see below] is more complicated). If η is both frequency- and amplitude-dependent, this is a can of worms! (What do we do for a bath when one degree-of-freedom is not necessarily weakly perturbed? No general procedure known). Important observation: All effects of linearly-responding environments on the motion of a system are encapsulated in a single function J(ω) π 2 i C 2 i m i ω i δ(ω ω i ). (29) The quantity J(ω) can often be read off from the equation of motion obeyed by the system in the classical limit. In particular, for linear frequency-independent friction described by a term η q in the equation of motion, J(ω) = ηω (3) 5

6 In the general case of a shunting black box described by an admittance Y (ω), we have (AJL, Phys. Rev. B 3, 128 (1984)) J(ω) = Im (iωy (ω)) Imω (31) 7. Calculations based on the oscillator-bath model Any number of possibilities: e.g. to obtain classical damped equation of motion, simply write down the equations of motion for U and eliminate E (i.e., the oscillator bath). Most flexible is probably the Feynman-Vernon functional integral technique (applicable to real-time and imaginary-time (i.e. tunneling) problems). E.g. for simple ohmic dissipation the action which goes in the imaginary-time functional integral is simply S eff {q(τ)} = dτ { } 1 2 m q2 + V (q) + η 4π so the WKB exponent S eff / is always increased by the dissipation. dτdτ (q(τ) q(τ )) 2 (τ τ ) 2 (32) Other results: spin-boson model, more general two-state systems (T 1,T 2 ) Further references: C. W. Gardiner, Quantum Noise Springer, Berlin, 1991 N. G. van Kampen, Stochastic processes in physics and chemistry, 3 rd edition, Elsevier, 27 A. J. Leggett, Lecture notes for Statisitical Physics 26, especially lectures 2-24: 6