Dark soliton in a disorder potential Magorzata Mochol , Marcin - PowerPoint PPT Presentation

Introduction Classical description: Deformation of a dark soliton Quantum description Conclusions Dark soliton in a disorder potential Małgorzata Mochol , Marcin Płodzień, Krzysztof Sacha Institute of Physics Jagiellonian University in Cracow 13 th September 2012 M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 1/16

Introduction Classical description: Deformation of a dark soliton Quantum description Conclusions Introduction 1 Classical description: Deformation of a dark soliton 2 Expansion in Bogoliubov modes Expansion in modes of the Pöschl–Teller potential Comparision with numerical calculations Quantum description 3 Effective Hamiltonian Anderson localization of a dark soliton Conclusions 4 M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 2/16

Introduction Classical description: Deformation of a dark soliton Quantum description Conclusions We consider N 0 bosonic atoms in a 1D box potential of length L at zero temperature. Single particle state φ 0 is a solution of the Gross-Pitaevskii equation (GPE) − � 2 2 m ∂ 2 x φ 0 + g 0 | φ 0 | 2 φ 0 = µ 0 φ 0 , (1) There exist stationary solutions: dark soliton ( g 0 > 0) bright soliton ( g 0 < 0) � e − i θ � x − q � N φ 0 ( x − q ) = e − i θ √ ρ 0 tanh φ 0 ( x − q ) = � , . � 2 ξ ξ x − q cosh ξ by Marc Haelterman t = � . M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 3/16 E = µ , l = ξ,

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions In our considerations dark soliton is placed in a weak external potential V ( x ) . To calculate a small perturbation of solitonic wavefunction we start with time–independent GPE: − 1 x φ ( x ) + 1 2 ∂ 2 ρ 0 | φ ( x ) | 2 φ ( x ) + V ( x ) φ ( x ) = µφ ( x ) , (2) ↑ ↑ φ = φ 0 + δφ µ = µ 0 + δµ = 1 + δµ . ↓ Time–independent, non–homogeneous Bogoliubov–de Gennes equations: � δφ � − φ 0 � � � � φ 0 L = V + δµ , (3) δφ ∗ φ ∗ − φ ∗ 0 0 where � − 1 ρ 0 | φ 0 | 2 − 1 2 ∂ 2 2 + 1 ρ 0 φ 2 x + � 0 L = . (4) ρ 0 | φ 0 | 2 + 1 − 1 ρ 0 φ ∗ 2 1 2 ∂ 2 2 x − 0 M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 4/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions φ 0 ( x − q ) = e − i θ √ ρ 0 ξ tanh ( x − q ) . J. Dziarmaga, Phys. Rev. A 70 , 063616 (2004) Now we have all vectors to build a complete basis and deformation of the soliton can be expanded in that basis � δφ � u θ � u ad � u q � u ad � � � � � q θ = ∆ θ + P θ + ∆ q + P q δφ ∗ v ad v ad v θ v q θ q � u k � v ∗ � � �� � + b ∗ k + b k . (5) k u ∗ v k k k P θ M θ = − 2 � ∂ N 0 φ 0 | V φ 0 � + δµ − iR ( � u q | V φ 0 � + � v q | V φ ∗ 0 � ) = 0 ւ ց � L dx | φ ( x − q ) | 2 ∂ x V ( x ) = 0 δµ = 2 � ∂ N 0 φ 0 | V φ 0 � 0 ↓ � L δµ = 1 tanh y + y sech 2 y � � dy tanh yV ( y + q ) L 0 M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 5/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions Let us start again with the stationary GPE but assume the solution we are looking for is a real function � � − 1 x + 1 ρ 0 φ 2 − µ + V ( x ) 2 ∂ 2 φ = 0 . (6) ր տ µ = µ 0 + δµ = 1 + δµ φ = φ 0 + δφ φ 2 0 ( x − q ) = ρ 0 tanh 2 ( x − q ) = ρ 0 1 − cosh − 2 ( x − q ) � � ↓ x → x + q ( H 0 + 2 ) δφ = δµφ 0 − V ( x + q ) φ 0 , (7) where H 0 = − 1 2 ∂ 2 3 x − cosh 2 ( x ) – Pöschl-Teller Hamiltonian. M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 6/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions There are two bound states: J. Lekner, Am. J. Phys. 75 , 1151 (2007) ψ 0 ( x ) ∼ sech 2 ( x ) ∼ ∂ x φ 0 E 0 = − 2 E 1 = − 1 ψ 1 ( x ) ∼ sech ( x ) tanh ( x ) 2 and scattering states E k = k 2 ψ k ( x ) 2 We can therefore expand deformation δφ over orthonormal basis of eigenfunctions � δφ = α 0 ψ 0 + α 1 ψ 1 + dk α k ψ k ( x ) , (8) � dx ψ ∗ ( E j + 2 ) α j = j ( x )[ δµφ 0 − V ( x + q ) φ 0 ] . (9) M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 7/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions In order to solve ( H 0 + 2 ) δφ = δµφ 0 − V ( x + q ) φ 0 , (10) we have to invert the operator H 0 + 2 in the Hilbert space what is simple because all eigenfunctions of H 0 are known. That is � δφ ( x ) = dy K ( x , y ) [ δµφ 0 − V ( y + q ) φ 0 ] , (11) where the symmetric kernel K ( x , y ) reads � ψ k ( x ) ψ ∗ 2 k ( y ) 3 ψ 1 ( x ) ψ ∗ K ( x , y ) = 1 ( y ) + 2 4 + k 2 − 1 16 sech 2 ( x ) sech 2 ( y ) × = sh 2 2 x + sh 2 2 y + 4 ch 2 x + 4 ch 2 y � − 3 − ( ch 2 x + ch 2 y + 3 ) | sh 2 x − sh 2 y | − 4 sh | x − y | sh x sh y − 6 | x − y |} . (12) M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 8/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions In the present approach the chemical potential has to be determined by the normalization condition � φ | φ � = N 0 + O ( δφ 2 ) � dxdy φ 0 ( x ) K ( x , y ) V ( y + q ) φ 0 ( y ) = δµ � dxdy φ 0 ( x ) K ( x , y ) φ 0 ( y ) 1 � tanh y + y sech 2 y � � = dy tanh yV ( q + y ) , L (13) M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 9/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions Expansion in Bogoliubov modes: � [ b k u k ( x ) + b ∗ k v ∗ φ ( x ) = φ 0 ( x ) + k ( x )] (14) k Expansion in modes of the Pöschl–Teller potential: � dy K ( x , y ) V ( y + q ) φ 0 ( y ) + δµ ∂φ 0 ( x ) | µ 0 φ ( x ) = φ 0 ( x ) − , (15) ∂µ 0 M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 10/16

Introduction Expansion in Bogoliubov modes Classical description: Deformation of a dark soliton Expansion in modes of the Pöschl–Teller potential Quantum description Comparision with numerical calculations Conclusions 10 2 (b) (a) 1 V(x) 5 0 0 (c) (d) 1 1 wavefunction 0 0 -1 -1 -20 -10 0 10 20 -20 -10 0 10 20 x x Figure: In panel (a) we show an example of the optical speckle potential with the correlation length σ R = 0 . 05 and for V 0 = 1 while in panel (c) we present the corresponding solution of the Gross-Pitaevskii equation obtained numerically (solid black line) and within the perturbation approach (red dashed line). In panels (b) and (d) we show the same as in (a) and (c) but for σ R = 1 and V 0 = 0 . 5. Green dotted lines in (c) and (d) correspond to unperturbed soliton wavefunctions. M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 11/16

Introduction Classical description: Deformation of a dark soliton Effective Hamiltonian Quantum description Anderson localization of a dark soliton Conclusions Perturbative approach: � φ � φ 0 � u θ � u ad � u q � u ad � � � � � � q θ = + ∆ θ + P θ + ∆ q + P q φ ∗ φ ∗ v ad v ad v θ v q 0 θ q � u k � v ∗ � � �� � + b ∗ k + b k (16) . k u ∗ v k k k Non-perturbative Dziarmaga approach: J. Dziarmaga, Phys. Rev. A 70 , 063616 (2004) � φ � φ 0 � u ad � u ad � � � � θ q = + P θ + P q φ ∗ φ ∗ v ad v ad 0 q � u k θ � v ∗ � � �� � + b ∗ k + b k . (17) k u ∗ v k k k ↓ substitute � 1 � � 1 2 | ∂ x φ | 2 + V | φ | 2 + 2 ρ 0 | φ | 4 − µ | φ | 2 H = dx (18) + apply the second quantization formalism M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 12/16

Introduction Classical description: Deformation of a dark soliton Effective Hamiltonian Quantum description Anderson localization of a dark soliton Conclusions The quantum effective Hamiltonian: H = ˆ ˆ H q + ˆ H B + ˆ H 1 , (19) where ˆ P 2 � ˆ q dxV ( x ) | φ 0 ( x − q ) | 2 H q = − 2 | M q | + � ˆ � P 2 2 | M q | + | M q | � V ( x ) q = − dx , (20) cosh 2 ( x − q ) 4 ˆ � ǫ k ˆ k ˆ b † H B = b k , (21) k 0 � )(ˆ b k + ˆ ˆ � ( � u k | V φ 0 � + � v k | V φ ∗ b † H 1 = k ) . (22) k M. Mochol, M. Płodzień, K. Sacha Dark soliton in a disorder potential 13/16