Black hole instabilities. 2. Super-radiance A rotating Black Hole (Kerr) is subject to a weak instability : Classical waves display a super-radiance : φ in ω, l , m → R ω, l , m φ out ω, l , m + T ω, l , m φ absorbed , ω, l , m with | R ω, l , m | 2 = 1 + | T ω, l , m | 2 > 1 . Energy is extracted from the hole. This is a stimulated process. At the Quantum level, super-radiance implies a spontaneous pair creation process, i.e. a "vacuum instability" , decay rate ∝ | T ω, l , m | 2 Unruh and Starobinski (1973) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 3. Black hole Bomb When introducing a reflecting boundary condition , the super-radiance induces a dynamical instability, a Black Hole Bomb , Press ’70, Kang ’97, Cardoso et al ’04. A non-zero mass can induce the reflection , Damour et al ’76 . Could be used to constrain the mass of hypothetical axions . As in a resonant cavity, the spectrum now contains a discrete set of modes with complex frequencies. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 3. Black hole Bomb When introducing a reflecting boundary condition , the super-radiance induces a dynamical instability, a Black Hole Bomb , Press ’70, Kang ’97, Cardoso et al ’04. A non-zero mass can induce the reflection , Damour et al ’76 . Could be used to constrain the mass of hypothetical axions . As in a resonant cavity, the spectrum now contains a discrete set of modes with complex frequencies. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 3. Black hole Bomb When introducing a reflecting boundary condition , the super-radiance induces a dynamical instability, a Black Hole Bomb , Press ’70, Kang ’97, Cardoso et al ’04. A non-zero mass can induce the reflection , Damour et al ’76 . Could be used to constrain the mass of hypothetical axions . As in a resonant cavity, the spectrum now contains a discrete set of modes with complex frequencies. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 4. Hawking radiation In 1974, Hawking showed that a Schwarzschild Black Hole spontaneously emits thermal radiation. Even though it is micro-canonically stable , it is canonically unstable : C v < 0. In fact, the partition function possesses one unstable bound mode (Gross-Perry-Yaffe ’82). The same bound mode is responsible for the dynamical instability of 5 dimensional Black String (Gregory-Laflamme ’93). Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 4. Hawking radiation In 1974, Hawking showed that a Schwarzschild Black Hole spontaneously emits thermal radiation. Even though it is micro-canonically stable , it is canonically unstable : C v < 0. In fact, the partition function possesses one unstable bound mode (Gross-Perry-Yaffe ’82). The same bound mode is responsible for the dynamical instability of 5 dimensional Black String (Gregory-Laflamme ’93). Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 4. Hawking radiation In 1974, Hawking showed that a Schwarzschild Black Hole spontaneously emits thermal radiation. Even though it is micro-canonically stable , it is canonically unstable : C v < 0. In fact, the partition function possesses one unstable bound mode (Gross-Perry-Yaffe ’82). The same bound mode is responsible for the dynamical instability of 5 dimensional Black String (Gregory-Laflamme ’93). Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 5. Black Hole Lasers discovered by Corley & Jacobson in 1999, arises in the presence of two horizons (charged BH) and dispersion , either superluminal or subluminal , the ’trapped’ region acts as a cavity , induces an exponential growth of Hawking radiation , and this is a dynamical instability . Naturally arises in supersonic flows in BEC → no hypothesis → experiments ? (Technion, June 2009) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 5. Black Hole Lasers discovered by Corley & Jacobson in 1999, arises in the presence of two horizons (charged BH) and dispersion , either superluminal or subluminal , the ’trapped’ region acts as a cavity , induces an exponential growth of Hawking radiation , and this is a dynamical instability . Naturally arises in supersonic flows in BEC → no hypothesis → experiments ? (Technion, June 2009) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole instabilities. 5. Black Hole Lasers discovered by Corley & Jacobson in 1999, arises in the presence of two horizons (charged BH) and dispersion , either superluminal or subluminal , the ’trapped’ region acts as a cavity , induces an exponential growth of Hawking radiation , and this is a dynamical instability . Naturally arises in supersonic flows in BEC → no hypothesis → experiments ? (Technion, June 2009) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Bose Einstein Condensates Set of atoms is described by ˆ Ψ( t , x ) obeying [ˆ Ψ( t , x ) , ˆ Ψ † ( t , x ′ )] = δ 3 ( x − x ′ ) , and by a Hamiltonian � � 2 � Ψ + g ( x ) � Ψ † ∇ x ˆ Ψ † ˆ Ψ † ˆ Ψ † ˆ ˆ 2 m ∇ x ˆ Ψ + V ( x ) ˆ ˆ Ψˆ d 3 x H = Ψ . 2 at low temperature, condensation ˆ Ψ 0 ( t , x ) + ˆ Ψ( t , x ) = ψ ( t , x ) Ψ 0 ( t , x ) ( 1 + ˆ = φ ( t , x )) , (1) Ψ 0 ( t , x ) describes the condensed atoms , ˆ φ ( t , x ) describes relative perturbations . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Elongated 1D statio condensates A 1D stationary condensate is described by Ψ 0 ( t , x ) = e − i µ t / � × � ρ 0 ( x ) e i θ 0 ( x ) , ρ 0 is the mean density and v = � m ∂ x θ 0 the mean velocity . ρ 0 , v are determined by V and g through the Gross Pitaevskii eq. √ ρ 0 ∂ 2 2 mv 2 − � 2 µ = 1 x + V ( x ) + g ( x ) ρ 0 , 2 m ρ 0 which also gives ∂ x ( v ρ 0 ) = 0 . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
BdG equation for relative density fluctuations In a BEC, density fluctuations obey the BdG equation . In non-homogeneous condensates, it is appropriate to use relative density fluctuations which obey reads � T v + mc 2 � φ + mc 2 ˆ i � ( ∂ t + v ∂ x ) ˆ ˆ φ † , φ = (2) c 2 ( x ) ≡ g ( x ) ρ 0 ( x ) , m is the x -dep. speed of sound and T v a kinetic term T v ≡ − � 2 1 2 m v ( x ) ∂ x v ( x ) ∂ x . Only v ( x ) and c ( x ) enter in BdG eq. An exact result . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
BdG equation for relative density fluctuations In a BEC, density fluctuations obey the BdG equation . In non-homogeneous condensates, it is appropriate to use relative density fluctuations which obey reads � T v + mc 2 � φ + mc 2 ˆ i � ( ∂ t + v ∂ x ) ˆ ˆ φ † , φ = (2) c 2 ( x ) ≡ g ( x ) ρ 0 ( x ) , m is the x -dep. speed of sound and T v a kinetic term T v ≡ − � 2 1 2 m v ( x ) ∂ x v ( x ) ∂ x . Only v ( x ) and c ( x ) enter in BdG eq. An exact result . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
BdG equation for relative density fluctuations In a BEC, density fluctuations obey the BdG equation . In non-homogeneous condensates, it is appropriate to use relative density fluctuations which obey reads � T v + mc 2 � φ + mc 2 ˆ i � ( ∂ t + v ∂ x ) ˆ ˆ φ † , φ = (2) c 2 ( x ) ≡ g ( x ) ρ 0 ( x ) , m is the x -dep. speed of sound and T v a kinetic term T v ≡ − � 2 1 2 m v ( x ) ∂ x v ( x ) ∂ x . Only v ( x ) and c ( x ) enter in BdG eq. An exact result . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Link with extended Gravity, a side remark Because phonons only see the macrosc. mean fields c ( x ) , v ( x ) , ρ 0 ( x ) , are insensitive to microsc. qtts g ( x ) , V ( x ) and Q.pot. one can forget about the (fundamental) theory of the condensate when computing the phonon spectrum . → consider the BdG eq. from a 4D point of view by introducing 4D tensors the acoustic metric g µν ( t , x ) Unruh ’81 (hydrodyn. limit) a unit time-like vector field u µ ( t , x ) Jacobson ’96 (to implement locally dispersion.) extra scalars ... make link with Horava-Jacobson extended Gravity Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Link with extended Gravity, a side remark Because phonons only see the macrosc. mean fields c ( x ) , v ( x ) , ρ 0 ( x ) , are insensitive to microsc. qtts g ( x ) , V ( x ) and Q.pot. one can forget about the (fundamental) theory of the condensate when computing the phonon spectrum . → consider the BdG eq. from a 4D point of view by introducing 4D tensors the acoustic metric g µν ( t , x ) Unruh ’81 (hydrodyn. limit) a unit time-like vector field u µ ( t , x ) Jacobson ’96 (to implement locally dispersion.) extra scalars ... make link with Horava-Jacobson extended Gravity Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Link with extended Gravity, a side remark Because phonons only see the macrosc. mean fields c ( x ) , v ( x ) , ρ 0 ( x ) , are insensitive to microsc. qtts g ( x ) , V ( x ) and Q.pot. one can forget about the (fundamental) theory of the condensate when computing the phonon spectrum . → consider the BdG eq. from a 4D point of view by introducing 4D tensors the acoustic metric g µν ( t , x ) Unruh ’81 (hydrodyn. limit) a unit time-like vector field u µ ( t , x ) Jacobson ’96 (to implement locally dispersion.) extra scalars ... make link with Horava-Jacobson extended Gravity Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Link with extended Gravity, a side remark Because phonons only see the macrosc. mean fields c ( x ) , v ( x ) , ρ 0 ( x ) , are insensitive to microsc. qtts g ( x ) , V ( x ) and Q.pot. one can forget about the (fundamental) theory of the condensate when computing the phonon spectrum . → consider the BdG eq. from a 4D point of view by introducing 4D tensors the acoustic metric g µν ( t , x ) Unruh ’81 (hydrodyn. limit) a unit time-like vector field u µ ( t , x ) Jacobson ’96 (to implement locally dispersion.) extra scalars ... make link with Horava-Jacobson extended Gravity Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Link with extended Gravity, a side remark Because phonons only see the macrosc. mean fields c ( x ) , v ( x ) , ρ 0 ( x ) , are insensitive to microsc. qtts g ( x ) , V ( x ) and Q.pot. one can forget about the (fundamental) theory of the condensate when computing the phonon spectrum . → consider the BdG eq. from a 4D point of view by introducing 4D tensors the acoustic metric g µν ( t , x ) Unruh ’81 (hydrodyn. limit) a unit time-like vector field u µ ( t , x ) Jacobson ’96 (to implement locally dispersion.) extra scalars ... make link with Horava-Jacobson extended Gravity Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing phonon spectra. Three objectives: A. Determine the structure of real and complex eigen-frequency modes. B. Compute the discrete set of complex frequencies. C. Understand the link between the BH laser spectrum and Hawking radiation. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing phonon spectra. Three objectives: A. Determine the structure of real and complex eigen-frequency modes. B. Compute the discrete set of complex frequencies. C. Understand the link between the BH laser spectrum and Hawking radiation. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing phonon spectra. basically equivalent to that of a hermitian scalar field . to handle the complex character of ˆ φ , introduce the doublet (Leonhardt et al. ’03) ˆ � � φ ˆ W ≡ , ˆ φ † invariant under the pseudo-Hermitian conjugation ( pH . c . ) W = ¯ ˆ W ≡ σ 1 ˆ ˆ W † . Thus, the mode decomposition of ˆ W is ˆ � a n + ¯ a † � ( W n ˆ W n ˆ ( W n ˆ W = n ) = a n + pH . c . ) , (3) n n � u n � where W n ( t , x ) are doublets of C -functions: W n = . v n Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing phonon spectra. basically equivalent to that of a hermitian scalar field . to handle the complex character of ˆ φ , introduce the doublet (Leonhardt et al. ’03) ˆ � � φ ˆ W ≡ , ˆ φ † invariant under the pseudo-Hermitian conjugation ( pH . c . ) W = ¯ ˆ W ≡ σ 1 ˆ ˆ W † . Thus, the mode decomposition of ˆ W is ˆ � a n + ¯ a † � ( W n ˆ W n ˆ ( W n ˆ W = n ) = a n + pH . c . ) , (3) n n � u n � where W n ( t , x ) are doublets of C -functions: W n = . v n Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing phonon spectra. basically equivalent to that of a hermitian scalar field . to handle the complex character of ˆ φ , introduce the doublet (Leonhardt et al. ’03) ˆ � � φ ˆ W ≡ , ˆ φ † invariant under the pseudo-Hermitian conjugation ( pH . c . ) W = ¯ ˆ W ≡ σ 1 ˆ ˆ W † . Thus, the mode decomposition of ˆ W is ˆ � a n + ¯ a † � ( W n ˆ W n ˆ ( W n ˆ W = n ) = a n + pH . c . ) , (3) n n � u n � where W n ( t , x ) are doublets of C -functions: W n = . v n Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing spectra. The inner product The conserved inner product � d x ρ 0 ( x ) W ∗ � W 1 | W 2 � ≡ 1 ( t , x ) σ 3 W 2 ( t , x ) , (4) is not positive definite (c.f. the Klein-Gordon product). As usual , mode orthogonality � W n | W m � = −� ¯ W n | ¯ W m � = δ nm , and ETC imply canonical commutators a † [ˆ a n , ˆ m ] = δ nm , where a n = � W n | ˆ ˆ W � . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing spectra. The inner product The conserved inner product � d x ρ 0 ( x ) W ∗ � W 1 | W 2 � ≡ 1 ( t , x ) σ 3 W 2 ( t , x ) , (4) is not positive definite (c.f. the Klein-Gordon product). As usual , mode orthogonality � W n | W m � = −� ¯ W n | ¯ W m � = δ nm , and ETC imply canonical commutators a † [ˆ a n , ˆ m ] = δ nm , where a n = � W n | ˆ ˆ W � . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing spectra. The inner product The conserved inner product � d x ρ 0 ( x ) W ∗ � W 1 | W 2 � ≡ 1 ( t , x ) σ 3 W 2 ( t , x ) , (4) is not positive definite (c.f. the Klein-Gordon product). As usual , mode orthogonality � W n | W m � = −� ¯ W n | ¯ W m � = δ nm , and ETC imply canonical commutators a † [ˆ a n , ˆ m ] = δ nm , where a n = � W n | ˆ ˆ W � . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The notion of Asympt. Bounded Modes For stationary condensates with infinite spatial extension the solutions of H W λ ( x ) = λ W λ ( x ) , (5) which belong to the spectrum must be Asymptotically Bounded : bounded for x → ±∞ . Since the scalar product is non-positive def. the frequency λ can be complex in non-homog. cond. Since Quasi Normal Modes are not ABM , they are not in the spectrum. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The notion of Asympt. Bounded Modes For stationary condensates with infinite spatial extension the solutions of H W λ ( x ) = λ W λ ( x ) , (5) which belong to the spectrum must be Asymptotically Bounded : bounded for x → ±∞ . Since the scalar product is non-positive def. the frequency λ can be complex in non-homog. cond. Since Quasi Normal Modes are not ABM , they are not in the spectrum. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The notion of Asympt. Bounded Modes For stationary condensates with infinite spatial extension the solutions of H W λ ( x ) = λ W λ ( x ) , (5) which belong to the spectrum must be Asymptotically Bounded : bounded for x → ±∞ . Since the scalar product is non-positive def. the frequency λ can be complex in non-homog. cond. Since Quasi Normal Modes are not ABM , they are not in the spectrum. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The notion of Asympt. Bounded Modes For stationary condensates with infinite spatial extension the solutions of H W λ ( x ) = λ W λ ( x ) , (5) which belong to the spectrum must be Asymptotically Bounded : bounded for x → ±∞ . Since the scalar product is non-positive def. the frequency λ can be complex in non-homog. cond. Since Quasi Normal Modes are not ABM , they are not in the spectrum. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons A sonic horizon is found at c ( x ) = | v ( x ) | . Take v < 0. If c , v are smooth , near the horizon, c ( x ) + v ( x ) ∼ κ x where κ = ∂ x ( c + v ) | hor . , decay rate ∼ "surface gravity" . Without dispersion, x ( t ) = x 0 e κ t and p ( t ) = p 0 e − κ t : standard near horiz. behav. ( p ( x ) : local wave number) and standard Hawking temperature: k B T H = � κ/ 2 π . With dispersion, x ( t ) � = x 0 e κ t but p ( t ) = p 0 e − κ t still found, This is the root of the robustness of the spectrum, , 2011 Como School lectures, and PRD 83 , 024021 (2012) . see RP Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons A sonic horizon is found at c ( x ) = | v ( x ) | . Take v < 0. If c , v are smooth , near the horizon, c ( x ) + v ( x ) ∼ κ x where κ = ∂ x ( c + v ) | hor . , decay rate ∼ "surface gravity" . Without dispersion, x ( t ) = x 0 e κ t and p ( t ) = p 0 e − κ t : standard near horiz. behav. ( p ( x ) : local wave number) and standard Hawking temperature: k B T H = � κ/ 2 π . With dispersion, x ( t ) � = x 0 e κ t but p ( t ) = p 0 e − κ t still found, This is the root of the robustness of the spectrum, , 2011 Como School lectures, and PRD 83 , 024021 (2012) . see RP Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons A sonic horizon is found at c ( x ) = | v ( x ) | . Take v < 0. If c , v are smooth , near the horizon, c ( x ) + v ( x ) ∼ κ x where κ = ∂ x ( c + v ) | hor . , decay rate ∼ "surface gravity" . Without dispersion, x ( t ) = x 0 e κ t and p ( t ) = p 0 e − κ t : standard near horiz. behav. ( p ( x ) : local wave number) and standard Hawking temperature: k B T H = � κ/ 2 π . With dispersion, x ( t ) � = x 0 e κ t but p ( t ) = p 0 e − κ t still found, This is the root of the robustness of the spectrum, , 2011 Como School lectures, and PRD 83 , 024021 (2012) . see RP Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons A sonic horizon is found at c ( x ) = | v ( x ) | . Take v < 0. If c , v are smooth , near the horizon, c ( x ) + v ( x ) ∼ κ x where κ = ∂ x ( c + v ) | hor . , decay rate ∼ "surface gravity" . Without dispersion, x ( t ) = x 0 e κ t and p ( t ) = p 0 e − κ t : standard near horiz. behav. ( p ( x ) : local wave number) and standard Hawking temperature: k B T H = � κ/ 2 π . With dispersion, x ( t ) � = x 0 e κ t but p ( t ) = p 0 e − κ t still found, This is the root of the robustness of the spectrum, , 2011 Como School lectures, and PRD 83 , 024021 (2012) . see RP Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons A sonic horizon is found at c ( x ) = | v ( x ) | . Take v < 0. If c , v are smooth , near the horizon, c ( x ) + v ( x ) ∼ κ x where κ = ∂ x ( c + v ) | hor . , decay rate ∼ "surface gravity" . Without dispersion, x ( t ) = x 0 e κ t and p ( t ) = p 0 e − κ t : standard near horiz. behav. ( p ( x ) : local wave number) and standard Hawking temperature: k B T H = � κ/ 2 π . With dispersion, x ( t ) � = x 0 e κ t but p ( t ) = p 0 e − κ t still found, This is the root of the robustness of the spectrum, , 2011 Como School lectures, and PRD 83 , 024021 (2012) . see RP Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons A sonic horizon is found at c ( x ) = | v ( x ) | . Take v < 0. If c , v are smooth , near the horizon, c ( x ) + v ( x ) ∼ κ x where κ = ∂ x ( c + v ) | hor . , decay rate ∼ "surface gravity" . Without dispersion, x ( t ) = x 0 e κ t and p ( t ) = p 0 e − κ t : standard near horiz. behav. ( p ( x ) : local wave number) and standard Hawking temperature: k B T H = � κ/ 2 π . With dispersion, x ( t ) � = x 0 e κ t but p ( t ) = p 0 e − κ t still found, This is the root of the robustness of the spectrum, , 2011 Como School lectures, and PRD 83 , 024021 (2012) . see RP Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons Our globally defined flow is � κ x � c ( x ) + v ( x ) = c H D tanh , c H D where D fixes the asymp. value of c + v = ± c H D Without dispersion, D plays no spectral role With dispersion, D is highly relevant . E.g., it fixes the cut-off frequency ω max ∼ c H ξ D 3 / 2 . where ξ is the healing length. In opt. fibers, D ≤ 10 − 3 ! Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons Our globally defined flow is � κ x � c ( x ) + v ( x ) = c H D tanh , c H D where D fixes the asymp. value of c + v = ± c H D Without dispersion, D plays no spectral role With dispersion, D is highly relevant . E.g., it fixes the cut-off frequency ω max ∼ c H ξ D 3 / 2 . where ξ is the healing length. In opt. fibers, D ≤ 10 − 3 ! Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons Our globally defined flow is � κ x � c ( x ) + v ( x ) = c H D tanh , c H D where D fixes the asymp. value of c + v = ± c H D Without dispersion, D plays no spectral role With dispersion, D is highly relevant . E.g., it fixes the cut-off frequency ω max ∼ c H ξ D 3 / 2 . where ξ is the healing length. In opt. fibers, D ≤ 10 − 3 ! Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Smooth sonic horizons Our globally defined flow is � κ x � c ( x ) + v ( x ) = c H D tanh , c H D where D fixes the asymp. value of c + v = ± c H D Without dispersion, D plays no spectral role With dispersion, D is highly relevant . E.g., it fixes the cut-off frequency ω max ∼ c H ξ D 3 / 2 . where ξ is the healing length. In opt. fibers, D ≤ 10 − 3 ! Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Spectrum of W n for one B/W sonic horizon The complete set of modes is Macher-RP 2009 a continuous set of real frequency modes which contains for ω > ω max , two positive norm modes, as in flat space, W u ω , W v ω , which resp. describe right/left moving phonons, for 0 < ω < ω max , three modes: 2 positive norm W u ω , W v ω + 1 negative norm mode ¯ W u − ω . The threashold freq. ω max scales 1 / healing length = mc / � . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Spectrum of W n for one B/W sonic horizon Lessons: There are no complex freq. ABM , Same spectrum for White Holes and Black Holes , because invariant under v → − v . Hence White Hole flows are dyn. stable , as BH ones. Yet, WH flows display specific features: "undulations" or "hydraulic jumps" (zero freq. modes with macroscopic amplitudes) Mayoral et al. and Vancouver experiment 2010 Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Spectrum of W n for one B/W sonic horizon Lessons: There are no complex freq. ABM , Same spectrum for White Holes and Black Holes , because invariant under v → − v . Hence White Hole flows are dyn. stable , as BH ones. Yet, WH flows display specific features: "undulations" or "hydraulic jumps" (zero freq. modes with macroscopic amplitudes) Mayoral et al. and Vancouver experiment 2010 Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The scattering of in -modes For ω > ω max , there is an elastic scattering: W u , in = T ω W u , out + R ω W v , out , with , ω ω ω | T ω | 2 + | R ω | 2 = 1 . For 0 < ω < ω max , there is a 3 × 3 matrix, e.g. + β ω ¯ W u , out W u , in = α ω W u , out + R ω W v , out , (6) ω ω ω − ω with | α ω | 2 + | R ω | 2 − | β ω | 2 = 1 . The β coefficients describe a super-radiance , hence a vacuum instability in QM, i.e. the spontaneous sonic B/W hole radiation. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The scattering of in -modes For ω > ω max , there is an elastic scattering: W u , in = T ω W u , out + R ω W v , out , with , ω ω ω | T ω | 2 + | R ω | 2 = 1 . For 0 < ω < ω max , there is a 3 × 3 matrix, e.g. + β ω ¯ W u , out W u , in = α ω W u , out + R ω W v , out , (6) ω ω ω − ω with | α ω | 2 + | R ω | 2 − | β ω | 2 = 1 . The β coefficients describe a super-radiance , hence a vacuum instability in QM, i.e. the spontaneous sonic B/W hole radiation. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The properties of the BH radiation For ω max ≥ 2 κ , the energy spectrum f ω = ω | β ω | 2 is JM-RP ’09 accurately Planckian (up to ω max ) and ( f ω = ω/ ( e ω/ T ω − 1 ) , with a temperature κ/ 2 π = T Hawking , exactly as predicted by the gravitational analogy . 0 -1 1.1 -2 log 10 f ω T ω / T H -3 1 D=0.1 D=0.4 D=0.7 -4 0.9 -5 -6 0.8 0.01 0.1 1 0.01 0.1 1 ω / κ ω / κ Spectra obtained from the BdG eq. only . Determine the validity domain of the Unruh analogy. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The properties of the BH radiation For ω max ≥ 2 κ , the energy spectrum f ω = ω | β ω | 2 is JM-RP ’09 accurately Planckian (up to ω max ) and ( f ω = ω/ ( e ω/ T ω − 1 ) , with a temperature κ/ 2 π = T Hawking , exactly as predicted by the gravitational analogy . 0 -1 1.1 -2 log 10 f ω T ω / T H -3 1 D=0.1 D=0.4 D=0.7 -4 0.9 -5 -6 0.8 0.01 0.1 1 0.01 0.1 1 ω / κ ω / κ Spectra obtained from the BdG eq. only . Determine the validity domain of the Unruh analogy. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Stationary profiles with 2 sonic horizons For two horizons: � κ W ( x + L ) � � κ B ( x − L ) � c ( x ) + v ( x ) = c H D tanh tanh . c H D c H D The distance between the 2 hor. is 2 L . 0,8 0,4 -12,5 -10 -7,5 -5 -2,5 0 2,5 5 7,5 10 12,5 -0,4 -0,8 -1,2 -1,6 -2 Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Black hole lasers in BEC, former works studied in terms of time-dep. wave-packets , both by Corley & Jacobson in ’99, and Leonhardt & Philbin in ’08. instead, in what follows, a spectral analysis of stationary modes . see also Garay et al. PRL 85 and PRA 63 (2000/1), BH/WH flows in BEC Barcelo et al. PRD 74 (2006), Dynam. stability analysis and Jain et al. PRA 76 (2007). Quantum De Laval nozzle Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Spectrum of W n for 2 sonic horizons The complete set of modes contains AC+RP 2010 a continuous spectrum of real freq. modes W u ω , W v ω with 0 < ω < ∞ , with positive norm only , and of dim. 2 . a discrete set of pairs of complex freq. modes ( V a , Z a ) with cc freq. ( λ a , λ ∗ a ) , with ℜ λ a ≤ ω max and a = 1 , .. N < ∞ . N.B. Negative norm modes ¯ W − ω are no longer in the spectrum; hence there is no Bogoliubov transformation in the present case. The field operator thus reads � ∞ � � ˆ � e − i ω t W α a α ω ( x ) ˆ W = d ω ω + pH . c . 0 α = u , v � � b a + e − i λ ∗ � e − i λ a t V a ( x ) ˆ a t Z a ( x ) ˆ + c a + pH . c . . (7) a Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Spectrum of W n for 2 sonic horizons The complete set of modes contains AC+RP 2010 a continuous spectrum of real freq. modes W u ω , W v ω with 0 < ω < ∞ , with positive norm only , and of dim. 2 . a discrete set of pairs of complex freq. modes ( V a , Z a ) with cc freq. ( λ a , λ ∗ a ) , with ℜ λ a ≤ ω max and a = 1 , .. N < ∞ . N.B. Negative norm modes ¯ W − ω are no longer in the spectrum; hence there is no Bogoliubov transformation in the present case. The field operator thus reads � ∞ � � ˆ � e − i ω t W α a α ω ( x ) ˆ W = d ω ω + pH . c . 0 α = u , v � � b a + e − i λ ∗ � e − i λ a t V a ( x ) ˆ a t Z a ( x ) ˆ + c a + pH . c . . (7) a Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Spectrum of W n for 2 sonic horizons The complete set of modes contains AC+RP 2010 a continuous spectrum of real freq. modes W u ω , W v ω with 0 < ω < ∞ , with positive norm only , and of dim. 2 . a discrete set of pairs of complex freq. modes ( V a , Z a ) with cc freq. ( λ a , λ ∗ a ) , with ℜ λ a ≤ ω max and a = 1 , .. N < ∞ . N.B. Negative norm modes ¯ W − ω are no longer in the spectrum; hence there is no Bogoliubov transformation in the present case. The field operator thus reads � ∞ � � ˆ � e − i ω t W α a α ω ( x ) ˆ W = d ω ω + pH . c . 0 α = u , v � � b a + e − i λ ∗ � e − i λ a t V a ( x ) ˆ a t Z a ( x ) ˆ + c a + pH . c . . (7) a Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Norms and commutators The real freq., the modes W α a α ω and operators ˆ ω obey ω | W α ′ W α ′ � W α ω ′ � = δ ( ω − ω ′ ) δ αα ′ = −� ¯ W α ω | ¯ ω ′ � and a α ′ † a α ω ′ ] = δ ( ω − ω ′ ) δ αα ′ . [ˆ ω , ˆ Instead for complex frequency λ a , one has � V a | V a ′ � = 0 = � Z a | Z a ′ � , � V a | Z a ′ � = i δ aa ′ , (8) and [ˆ b a , ˆ b † [ˆ c † b a , ˆ a ′ ] = 0 , a ′ ] = i δ aa ′ . (9) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The two-mode sectors with complex freq. λ a Each pair (ˆ b a , ˆ c a ) always describes one complex, rotating, unstable oscillator: Its (Hermitian) Hamiltonian is ˆ a ˆ c † H a = − i λ a ˆ b a + H . c . (10) Writing λ a = ω a + i Γ a , with ω a , Γ a real > 0, ℜ λ a = ω a fixes the angular velocity, ℑ λ a = Γ a fixes the growth rate . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing the discrete spectrum of ABM The method: A. use WKB waves to 1. decompose the exact modes, 2. obtain algebraic relations (valid beyond WKB ) between the R freq. W ω and the C freq. V a , Z a B. a numerical analysis to validate the predictions. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing the discrete spectrum of ABM The method: A. use WKB waves to 1. decompose the exact modes, 2. obtain algebraic relations (valid beyond WKB ) between the R freq. W ω and the C freq. V a , Z a B. a numerical analysis to validate the predictions. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The scattering of real freq. u -mode 0,8 0,4 -12,5 -10 -7,5 -5 -2,5 0 2,5 5 7,5 10 12,5 -0,4 -0,8 -1,2 -1,6 -2 On the left of the White hor. W u , in → W u ω , the WKB sol. ω Between the two horizons, for ω < ω max , W ( 1 ) W ( 2 ) W u , in ( x ) = A ω W u ω ( x ) + B ( 1 ) ¯ − ω ( x ) + B ( 2 ) ¯ − ω ( x ) , (11) ω ω ω On the right of the Black horizon, W u , in → e i θ ω W u ω . ω W ( i ) Negative norm/freq WKB modes ¯ − ω in (11). Hence " anomalous scattering" ( ∼ Bogoliubov transf.). fully described by the csts. A ω , B ( 1 ) ω , B ( 2 ) and θ ω . ω Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Relating A ω , B ( 1 ) ω , B ( 2 ) and θ ω to ω BH and WH Bogoliubov trsfs. ω , ¯ algebraically achieved by introd. a 2-vector ( W u W − ω ) , on which acts a 2 × 2 S -matrix Leonhardt 2008 this S -matrix can be decomposed as S = U 4 U 3 U 2 U 1 . where U 1 describes the scattering on the WH horizon . U 2 the propagation from the WH to the BH U 3 the scattering on the BH horizon . U 4 the escape to the right of W u ω and the return of ¯ W ( 2 ) − ω to the WH horizon. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Relating A ω , B ( 1 ) ω , B ( 2 ) and θ ω to ω BH and WH Bogoliubov trsfs. ω , ¯ algebraically achieved by introd. a 2-vector ( W u W − ω ) , on which acts a 2 × 2 S -matrix Leonhardt 2008 this S -matrix can be decomposed as S = U 4 U 3 U 2 U 1 . where U 1 describes the scattering on the WH horizon . U 2 the propagation from the WH to the BH U 3 the scattering on the BH horizon . U 4 the escape to the right of W u ω and the return of ¯ W ( 2 ) − ω to the WH horizon. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The four U matrices, (Leonhardt et al.) Explicitly, � α ω � e i S u � � 0 α ω z ω ω U 1 = S WH = , U 2 = , e − iS ( 1 ) α ω z ∗ ˜ α ω ˜ 0 − ω ω � γ ω � � 1 0 � γ ω w ω U 3 = S BH = , U 4 = , e i S ( 2 ) γ ω w ∗ ˜ γ ω ˜ 0 − ω ω where � L � R ω � � S ( i ) S u d x k u − k ( i ) ω ≡ ω ( x ) , − ω ≡ d x ω ( x ) , i = 1 , 2 , − L − L ω are H-Jacobi actions, and L ω and R ω are the two turning points. By unitarity, one has | α ω | 2 = | ˜ α ω | 2 , | α ω | 2 = 1 / ( 1 − | z ω | 2 ) . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The real freq. mode The mode W u , in ( x ) must be single-valued . ω Hence the coeff. B ( 2 ) of the trapped piece ω W ( 1 ) ¯ W ( 2 ) ¯ W u , in = A ω W u ω + B ( 1 ) − ω + B ( 2 ) ω ω ω − ω must obey � � � � e i θ ω 1 = S , B ( 2 ) B ( 2 ) ω ω which implies S 21 ( ω ) B ( 2 ) = 1 − S 22 ( ω ) . (12) ω The first key equation . (Valid beyond WKB.) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The real freq. mode The mode W u , in ( x ) must be single-valued . ω Hence the coeff. B ( 2 ) of the trapped piece ω W ( 1 ) ¯ W ( 2 ) ¯ W u , in = A ω W u ω + B ( 1 ) − ω + B ( 2 ) ω ω ω − ω must obey � � � � e i θ ω 1 = S , B ( 2 ) B ( 2 ) ω ω which implies S 21 ( ω ) B ( 2 ) = 1 − S 22 ( ω ) . (12) ω The first key equation . (Valid beyond WKB.) Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The complex frequency ABModes When Im λ = Γ > 0, → Im k u λ > 0, hence growth for x → −∞ . So any single-valued ABMode must satisfy � β a ( λ ) � 0 � � = S ( λ ) . (13) 1 1 This implies S 22 ( λ ) = 1 , β a = S 12 ( λ ) . (14) Second key result: The poles of B ( 2 ) = S 21 / ( 1 − S 22 ) correspond ω to the complex freq. λ a . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
ω | 2 as a function of ω real |B ( 2 ) Green dots are numerical values , the continuous red line is a sum of Lorentzians. Near a complex frequency λ a , solution of S 22 = 1, |B ( 2 ) ω | 2 ∼ C a / | ω − ω a − i Γ a | 2 , i.e. a Lorentzian. Above ω max no peaks, because no neg. norm WKB mode. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing the complex freq. λ a = ω a + i Γ a . The λ a ’s, are fixed by the cond. ABM + single-valued . Both conditions encoded in S 22 = 1. When the leaking-out amplitudes are small , | z ω | , | w ω | = | β ω /α ω | ≪ 1 , the supersonic region acts as a cavity : To zeroth order in z ω , w ω , S 22 = 1 fixes ℜ λ a = ω a by a Bohr-Sommerfeld condition � L S ( 1 ) − ω − S ( 2 ) dx [ − k ( 1 ) ω ( x ) + k ( 2 ) − ω + π = ω ( x )] + π = 2 π n , − L where n = 1 , 2 , ..., N . This explains the discreteness of the set. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing the complex freq. λ a = ω a + i Γ a . To second order in z ω , w ω , S 22 = 1 fixes Im λ a = Γ a to be ω a = | S 12 ( ω a ) | 2 = | z ω a + w ω a e i ψ a | 2 2 Γ a T b (15) T b ω a > 0 is the bounce time , given by ω = ∂ � S ( 2 ) − ω − S ( 1 ) � T b − ω + ” non HJ terms ” (16) ∂ω The phase in the cosine is ω a + S ( 1 ) ψ a = S u − ω a + other ” non HJ terms ” Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Computing the complex freq. λ a = ω a + i Γ a . To second order in z ω , w ω , S 22 = 1 fixes Im λ a = Γ a to be ω a = | S 12 ( ω a ) | 2 = | z ω a + w ω a e i ψ a | 2 2 Γ a T b (15) T b ω a > 0 is the bounce time , given by ω = ∂ � S ( 2 ) − ω − S ( 1 ) � T b − ω + ” non HJ terms ” (16) ∂ω The phase in the cosine is ω a + S ( 1 ) ψ a = S u − ω a + other ” non HJ terms ” Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Validity of theoretical predictions 2.0 1.5 ω | 2 |B ( 2 ) 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ω/ω max Dots are numerical values . The 22 red lines are the theo. predictions . Excellent agreement because Γ a /ω a ≪ 1. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The evolution of ω a and Γ a in terms of L . 1 0.006 0.9 0.005 0.8 0.004 0.7 ω a /ω max Γ a /κ 0.6 0.003 0.5 0.4 0.002 0.3 0.001 0.2 0.1 0 5 10 15 20 25 10 12 14 16 18 20 22 24 L κ L κ New bounded modes appear as L grows. The Γ a reach their maximal value for ω a /ω max ≪ 1. Γ a reach 0 because of (Young) interferences. The destruction is imperfect when z ω � = w ω . No bounded mode is destroyed as L grows. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
A typical growing mode with a high Γ a ( Γ /ω ∼ 1 / 20 ) 2.0 1.5 1.0 0.5 ℜ ϕ 4 ( x ) 0.0 � 0.5 � 1.0 � 1.5 � 100 � 50 0 50 100 κ x Highest amplitudes in the trapped region. Exponential decrease on the Right of the BH horizon. The spatial damping is proportional to the rate Γ a = Im λ a . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The pattern of density-density fluctuations � δρδρ � Different scales are used, the central square is the trapped region. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Physical predictions At late times w.r.t. the formation of the BH-WH, i.e. times ≫ 1 / Max Γ a , the mode with the highest Γ a dominates all observables . The classical and quantum descriptions coincide . At earlier times , if the in -state is (near) vacuum, the quantum settings must be used , and all complex freq. modes contribute to the observables At "early" times , i.e. ∆ t ≤ T Bounce Hawking radiation as if the WH were not present. the discreteness of the λ a -set is not yet visible, the resolution in ω being too small. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Physical predictions At late times w.r.t. the formation of the BH-WH, i.e. times ≫ 1 / Max Γ a , the mode with the highest Γ a dominates all observables . The classical and quantum descriptions coincide . At earlier times , if the in -state is (near) vacuum, the quantum settings must be used , and all complex freq. modes contribute to the observables At "early" times , i.e. ∆ t ≤ T Bounce Hawking radiation as if the WH were not present. the discreteness of the λ a -set is not yet visible, the resolution in ω being too small. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Physical predictions At late times w.r.t. the formation of the BH-WH, i.e. times ≫ 1 / Max Γ a , the mode with the highest Γ a dominates all observables . The classical and quantum descriptions coincide . At earlier times , if the in -state is (near) vacuum, the quantum settings must be used , and all complex freq. modes contribute to the observables At "early" times , i.e. ∆ t ≤ T Bounce Hawking radiation as if the WH were not present. the discreteness of the λ a -set is not yet visible, the resolution in ω being too small. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Physical predictions At late times w.r.t. the formation of the BH-WH, i.e. times ≫ 1 / Max Γ a , the mode with the highest Γ a dominates all observables . The classical and quantum descriptions coincide . At earlier times , if the in -state is (near) vacuum, the quantum settings must be used , and all complex freq. modes contribute to the observables At "early" times , i.e. ∆ t ≤ T Bounce Hawking radiation as if the WH were not present. the discreteness of the λ a -set is not yet visible, the resolution in ω being too small. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The quantum flux emitted by a BH-WH system, 1 1. A BH-WH system with 13 complex freq. modes. 0.01 � 2000 � 0.06 0.008 1500 0.05 � 1000 0.006 Γ a /κ d P / d T 0.04 ω a � T b 700 � 0.03 � 0.004 500 � 0.02 � 0.002 � � � � 0.01 300 � � � � � � � �� � � � � � � 0. 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ω/ω max ω/ω max Left: The 13 values of T Bounce (dots) and Γ a (squares) a Right: The continuous spectrum obtained without the WH vs. the corresponding discrete quantity for the BH-WH pair. Very different spectra in ω -space . Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The flux emitted by a BH-WH system, 2 Fluxes emitted after a finite lapse of time by a single BH (solid line) and the BH-WH pair (dashed). 10 300 P ( ω, T = 200 κ − 1 ) P ( ω, T = 30 κ − 1 ) 250 8 200 6 150 4 100 2 50 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ω/ω max ω/ω max Left : after ∆ t = 30 /κ , no sign yet of discreteness nor instab. the BH-WH pair emits Hawking-like radiation. Right : after ∆ t = 200 /κ , discreteness and instab. visible. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The flux emitted by a BH-WH system, 2 Fluxes emitted after a finite lapse of time by a single BH (solid line) and the BH-WH pair (dashed). 10 300 P ( ω, T = 200 κ − 1 ) P ( ω, T = 30 κ − 1 ) 250 8 200 6 150 4 100 2 50 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ω/ω max ω/ω max Left : after ∆ t = 30 /κ , no sign yet of discreteness nor instab. the BH-WH pair emits Hawking-like radiation. Right : after ∆ t = 200 /κ , discreteness and instab. visible. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The Technion BH-WH, June 2009, preliminary results 14 12 10 ω | 2 8 |B ( 2 ) 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 ω/ω max About 4 narrow unstable modes. Experiment too short by a factor of 10 to see the laser effect. Probably more than 4 complex freq. modes. Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The limit L → 0 When the distance 2 L suff. small , i.e. smaller than a composite critical scale d ξ = ξ 2 / 3 ( c H /κ ) 1 / 3 , no complex freq. modes, hence no dyn. instability, no radiation emitted, even though κ � = 0, no entanglement entropy. Useful limit to control the degree of instability in experiments ? Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
The limit L → 0 When the distance 2 L suff. small , i.e. smaller than a composite critical scale d ξ = ξ 2 / 3 ( c H /κ ) 1 / 3 , no complex freq. modes, hence no dyn. instability, no radiation emitted, even though κ � = 0, no entanglement entropy. Useful limit to control the degree of instability in experiments ? Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
Classical terms: Induced instability When sending a classical wave W in ( t , x ) , this induces the instability. N.B. It does it through the overlaps with the decaying modes Z a b a ≡ � Z a | W in � (17) which fix the amplitude of the growing mode V a : � � � e − i λ a t b a V a ( x ) + p . H . c . W in ( t , x ) → . (18) a Antonin Coutant, Stefano Finazzi, and Renaud Parentani Analogue black-holes in BECondensates
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