Quantum Brownian motion with nongaussian stochastic forces - PowerPoint PPT Presentation

Quantum Brownian motion with nongaussian stochastic forces Hing-Tong Cho Department of Physics, Tamkang University (Collaboration with Bei-Lok Hu) RQIN 2017 - YITP Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

Outline I. Quantum Brownian motion II. Nongaussian stochastic forces III. The Langevin equation and the master equation VI. Conclusions Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

I. Quantum Brownian motion A particle (system) coupled linearly to a set of harmonic oscillators (environment): � t � 1 � x 2 − V ( x ) S [ x ] = ds 2 M ˙ 0 � t � 1 n − 1 � � q 2 2 m n ω 2 n q 2 S e [ q n ] = 2 m n ˙ ds n 0 n � t � S int [ x , { q n } ] = ( − C n x q n ) ds 0 n (Schwinger, Feynman-Vernon, Caldeira-Leggett, Hu-Paz-Zhang, ...) Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The dynamics of the particle is governed by the CTP effective action e iS [ x + ] − iS [ x − ] × e i Γ[ x + , x − ] = � � � e iS e [ { q n + } ] − iS e [ { q n − } ] Dq n + Dq n − CTP n e iS int [ x + , { q n + } ] − iS int [ x − , { q n − } ] � e iS [ x + ] − iS [ x − ]+ iS IF [ x + , x − ] = where S IF is the influence action due to the quantum harmonic oscillators. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The influence action S IF can be expressed in terms of the Schwinger-Keldysh propagators S IF [ x + , x − ] 1 � � ds ds ′ = 2 n x + ( s ) G n ++ ( s , s ′ ) x + ( s ′ ) − x + ( s ) G n + − ( s , s ′ ) x − ( s ′ ) � − x − ( s ) G n − + ( s , s ′ ) x + ( s ′ ) + x − ( s ) G n −− ( s , s ′ ) x − ( s ′ ) � due to the corresponding boundary conditions. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The influence action S IF can be written as � t � s e iS IF e − i 0 ds 0 ds ′ [∆ x ( s ) η ( s − s ′ )Σ x ( s ′ )] = � t � t 0 ds ′ [∆ x ( s ) ν ( s − s ′ )∆ x ( s ′ )] e − 1 0 ds 2 where ∆ x ( s ) = x + ( s ) − x − ( s ) and Σ x ( s ) = x + ( s ) + x − ( s ), and C 2 η ( s − s ′ ) � η n ( s − s ′ ) = − � n sin ω n ( s − s ′ ) = 2 m n ω n n n C 2 � � ν ( s − s ′ ) ν n ( s − s ′ ) = n cos ω n ( s − s ′ ) = 2 m n ω n n n S IF is basically separated into its real and imaginary parts. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

Rewriting the imaginary part of S IF as e − 1 � ∆ x ν ∆ x 2 � D ξ e − 1 ξν − 1 ξ e − 1 � � ∆ x ν ∆ x = N 2 2 � D ξ e − 1 ( ξ − i ν ∆ x ) ν − 1 ( ξ − i ν ∆ x ) e − 1 � � ∆ x ν ∆ x = N 2 2 � � D ξ P [ ξ ] e i ξ ∆ x = N ξν − 1 ξ is the Gaussian probability density of the where P [ ξ ] = e − 1 � 2 stochastic force ξ . Due to this probability density one has the stochastic average � ξ ( s ) ξ ( s ′ ) � s = ν ( s − s ′ ) which is called the noise kernel. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

After this procedure the effective action Γ[ x + , x − ] = S [ x + ] − S [ x − ] � t � s ds ′ ∆ x ( s ) η ( s − s ′ )Σ x ( s ′ ) − ds 0 0 � t + ds ∆ x ( s ) ξ ( s ) 0 The equation of motion for the particle is then given by � δ Γ[ x + , x − ] � = 0 � δ x + � x + = x − = x Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The equation of motion is a Langevin equation with the stochastic force ξ ( t ), � t x + V ′ ( x ) + M ¨ ds η ( t − s ) x ( s ) = ξ ( t ) 0 The integral term is related to dissipation as one can write C 2 η ( t ) = d � n dt γ ( t ) ⇒ γ ( t ) = cos ω n t 2 m n ω 2 n n and we have � t x + V ′ ( x ) + M ¨ ds γ ( t − s ) ˙ x ( s ) = ξ ( t ) 0 η ( s − s ′ ) is called the dissipation kernel. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The dissipation kernel and the noise kernel are respectively the real and the imaginary parts of the same Green’s function. They are related by the fluctuation-dissipation relation (FDR) � ∞ ds ′ K ( s − s ′ ) γ ( s ′ ) ν ( s ) = −∞ where in this simple case � ∞ d ω K ( s ) = π ω cos ω s 0 Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

II. Nongaussian stochastic forces The Brownian motion model with a different interaction term, � t � 1 � x 2 − V ( x ) S [ x ] = ds 2 M ˙ 0 � t � 1 n − 1 � � q 2 2 m n ω 2 n q 2 S e [ q n ] = ds 2 m n ˙ n 0 n � t � − λ C n xq 2 � � S int [ x , { q n } ] = ds n 0 n Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The influence action can be expanded as a power series in λ . � δ A ( i ) [ x + , x − ] S IF [ x + , x − ] = i The first term gives � t � t � � � � δ A (1) = � � ds − δ V n ( x + ) − ds − δ V n ( x − ) 0 0 n n where δ V n ( x ) = λ C n x 2 m n ω n This term can be interpreted as a renormalization of the potential. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The second term reads � t � s ds ′ ∆( s )Σ( s ′ ) η ( s − s ′ ) δ A (2) = − ds 0 0 � t � t ds ′ ∆( s )∆( s ′ ) ν ( s − s ′ ) + i ds 0 0 with ∆( s ) ≡ x + ( s ) − x − ( s ) and Σ( s ) ≡ x + ( s ) + x − ( s ). Similarly to the bilinear interaction case, η and ν are related to the dissipation and the noise kernels respectively. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The third term is δ A (3) � t � s � s ′ ds ′′ ∆( s )Σ( s ′ )Σ( s ′′ ) × ds ′ = ds 0 0 0 � 8 λ 3 C 3 ν n ( s ′ − s ′′ ) η n ( s ′′ − s ) − η n ( s ′ − s ′′ ) ν n ( s ′′ − s ) n η n ( s − s ′ ) � � n � t � s � s ′ ds ′′ ∆( s )Σ( s ′ )∆( s ′′ ) × ds ′ + i ds 0 0 0 � 8 λ 3 C 3 ν n ( s ′ − s ′′ ) ν n ( s ′′ − s ) + η n ( s ′ − s ′′ ) η n ( s ′′ − s ) n η n ( s − s ′ ) � � n � t � s � s ′ ds ′′ ∆( s )∆( s ′ )Σ( s ′′ ) × ds ′ − i ds 0 0 0 � 8 λ 3 C 3 ν n ( s ′ − s ′′ ) η n ( s ′′ − s ) − η n ( s ′ − s ′′ ) ν n ( s ′′ − s ) n ν n ( s − s ′ ) � � n � t � s � s ′ ds ′′ ∆( s )∆( s ′ )∆( s ′′ ) × ds ′ + ds 0 0 0 � 8 λ 3 C 3 ν n ( s ′ − s ′′ ) ν n ( s ′′ − s ) + η n ( s ′ − s ′′ ) η n ( s ′′ − s ) n ν n ( s − s ′ ) � � n Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

To the next order with terms ∆ΣΣΣ, i ∆∆ΣΣ, ∆∆∆Σ, i ∆∆∆∆, and so on. Γ[ x + , x − ] � t � t = S [ x + ] − S [ x − ] − ds δ V ( x + ) + ds δ V ( x − ) 0 0 � t � t � t ds ∆( s ) H ( s ; Σ) + i ds ′ ∆( s )∆( s ′ ) N 2 ( s , s ′ ; Σ) − ds 2 0 0 0 � t � t � t − 1 ds ′′ ∆( s )∆( s ′ )∆( s ′′ ) N 3 ( s , s ′ , s ′′ ) + · · · ds ′ ds 3! 0 0 0 where the kernels H (0) ( s ; Σ) + H (1) ( s ; Σ) + · · · H ( s ; Σ) = N (0) 2 ( s , s ′ ) + N (1) N 2 ( s , s ′ ; Σ) = 2 ( s , s ′ ; Σ) + · · · N (1) N 3 ( s , s ′ , s ′′ ) 3 ( s , s ′ , s ′′ ) + · · · = Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

Main idea: Terms quadratic or higher in powers of ∆( s ) can be interpreted as the effect of a single stochastic force ξ ( s ). � t � t � t � t 0 ds ′ � t 0 ds ′ ∆( s )∆( s ′ ) N 2 ( s , s ′ ;Σ) − 1 0 ds ′′ ∆( s )∆( s ′ )∆( s ′′ ) N 3 ( s , s ′ , s ′′ ) ] e i [ i 0 ds 0 ds 2 3! � � t D ξ P [ ξ ] e i 0 ds ∆( s ) ξ ( s ) = where P [ ξ ] is the probability density of the stochastic force ξ ( s ). Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

The two-point correlation ξ ( s ) ξ ( s ′ ) = N 2 ( s , s ′ ; Σ) � � N 2 ( s , s ′ ; Σ) is the new noise kernel which is history dependent. The stochastic force is nongaussian. ξ ( s ) ξ ( s ′ ) ξ ( s ′′ ) = N 3 ( s , s ′ , s ′′ ) � � The probability density P [ ξ ] is also not gaussian. Possible application to the nongaussianity of the CMB anisotropy spectrum. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

Term proportional to ∆( s ) is related to the dissipation kernel γ . � s ds ′ γ ( s , s ′ ; Σ) ˙ H ( s , s ′ ; Σ) = Σ( s ′ ) 0 where γ ( s , s ′ ; Σ) = γ (0) ( s , s ′ ) + γ (1) ( s , s ′ ; Σ) + · · · γ ( s , s ′ ; Σ) should be viewed as the new dissipation kernel which is also history dependent. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces

Fluctuation-dissipation relation � ∞ N 2 ( s , s ′ ; Σ) = ds 1 K ( s , s 1 ; Σ) γ ( s 1 , s ′ ; Σ) −∞ where the fluctuation-dissipation kernel K ( s , s ′ ; Σ) will also be history dependent in general. Hing-Tong Cho Quantum Brownian motion with nongaussian stochastic forces