Hawking Radiation in Acoustic Black-Holes on an Ion Ring

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1 Hawking Radiation in Acoustic Black-Holes on an Ion Ring Benni Reznik In collaboration with, B. Horstman, S. Fagnocchi, J. I. Cirac Towards the Observation of Hawking Radiation in Condensed Matter Systems. IFIC, Valencia, February 1-7, 2009

2 Outline: 1. A discrete analogue model of a sonic hole toy model on a ion ring geometry. -- motivation, trapped ions. --numerical classical simulation of a BH. 2. Numerical evidence of Hawking radiation: -- backward evolution and Bogolibouv particle creation. --mode conversion (Unruh s sonic-hole) --Bloch oscillations (Corley and Jacobson, falling lattice ) --Dynamical creation and point-to-point correlations. `

3 Sonic dumb-hole Unruh 81, 95 Can we discretize the model? Corley & Jacobson 98 Jacobson & Mattingly 99 Lattice black holes.

4 Ion lattices Aarhus, Denmark: 40 Ca + (red) and 24 Mg + (blue) xford, England: 40 Ca + Innsbruck, Austria: 40 Ca + Boulder, USA: Hg + (mercury)

5 Trapped Ions Courtesy of R. Blatt

6 Trapped ions t 10ns e t sec m ν x 2 g 1/2 = 2mν 11nm T Seperation between ions: d ν = k B 5µ m 50µ K Size of the wave packet << wavelength of visible light.

7 Entanglement entropy in Field Theory A B Entanglement Area Could it be that Entanglement is the quantum source of BH entropy? Bombelli, Koul, Lee, Sorkin 86

8 Entanglement entropy 2-D Harmonic lattice Entanglement Area S AB = tr(ρ A log ρ A )

ν n a n a n A B H int =Ω(t)(e -iφ σ + (k) +e iφ σ - (k) )x k Paul

9 Simulating detection of vacuum entanglement A B Entanglement Entropy Ions internal levels H=H 0 +H int H 0 =ω z (σ za +σ zb )+ ν n a n a n A B H int =Ω(t)(e -iφ σ + (k) +e iφ σ - (k) )x k Paul Trap 1/ω z << T<<1/ν 0 A. Retzker, J. I. Cirac, B. Reznik, PRL, 2005.

10 Discrete BH analogue c Inhomogeneous, but stationary velocity profile v (θ). The necessary (de-)acceleration of the ions is guaranteed by a force F e on the ions. F e v velocity c(x) v(x) BLACK HOLE supersonic region F e Harmonic oscillations around the equilibrium motion are phonons with velocities c (θ) (v (θ)) 1/2. When v increases the sound velocity decreases and a Black and White horizons can form. x c v

11 Nature (1992), PRL (1992)

12 Ions on a ring N ions of mass m in a ring with radius L: H = NX i=1 4π 2 ~ 2 2mL 2 2 θ 2 i + NX V e (θ i )+V c (θ 1,...,θ N ) i=1 We treat small perturbations around the equilibrium motion θ i (t) =θ 0 i (t)+δθ i (t) and expand the Hamiltonian to second order in δθ i H = NX i=1 4π 2 ~ 2 2mL 2 2 δθ 2 i X f ij (t)δθ i δθ j (1) i6=j

13 Large N effective field limit For a slowly varying v (θ) the system Lagrangian for the scalar field Φ (θ 0 i (t),t)=δθ i (t) becomes L = ml2 (2π) 2 Z dθ n (θ) 2 h( t Φ + v (θ) θ Φ) 2 (id (θ, i θ ) Φ) 2i with D (θ,k) = c (θ) k + O (k 3 ), the density n (θ) = 1/(v (θ) T ). c (θ) = p 2(2π) 3 n (θ) e 2 /(4π² 0 ml 3 ). We have a position dependent dispersive fluid.

14 Creating a discrete BH We want the classical dumb-hole equilibrium motion: i+ t / T θi () t = gv( ) N where g maps the normalized indices i/n [0,1] monotonically increasing onto the angles θ [0,2π] and is periodically continued. The velocity profile is then v(θ) =g 0 g 1 (θ) /T

15 Dynamics t hˆξ i i t = X j G ij (t)hˆξ j i t t Γ(t) =G(t) Γ(t)+Γ(t) G(t)T. Note that since the initial state is Gaussian (either thermal or vacuum) The state will remain Gaussian for later times. Hence higher order Correlations can be computed via Wick s decomposition theorem.

16 Group velocity D (k) For only nearest neighbor interactions c =2e N. or c (θ) = p 2(2π) 3 n (θ) e 2 /(4π² 0 ml 3 )

17 Testing wave packet trajectories From Unruh s model we expect: -reflection at the horizon -Hawking s radiation thermal properties remain mostly unaffected -but in our case k has a maximal value.

18 Back in time 2 k 2πn ( ) 40π n δθ ( 0) = ke, n = 1,,20, k Outgoing wave (negative frequency) Values of δθ i () t during propagation backwarts in time. Here we have v = 0.83, k = 10, N = 1000, N2 = 1200, Incoming wave (Negative and positive frequency)

19 Scattering to Low k modes We find scattering to a low wavenumber negative frequency mode, on the second branch, which is mostly right moving. This scattering is consistent with the approximate Killing frequency conservation. It was argued that conformal invariance in 1D prevents scattering. We find that both for the full and truncated models there is a small nonvanishing scattering.

20 In the commoving frame Wave remains a right mover Unruh s process Scattering to a left mover Corley & Jacobson s Bloch oscillation

21 Late time outgoing and early time ingoing distributions x i (t) in the lab frame.

22 Positive and negative frequencies ω 1.0 D(k) ω 2 + branch k branch k D(k) -2 comoving frame Positivity is determined by the sign of D(k) ω is nearly conserved only if ka<1 Lab frame ω= k v + D(k) ω= k v - D(k)

23 Positive and negative modes If the phononic excitations are localized in the flat subsonic region, the excitations δθ i (t) =hδθ i i t and δ θ i (t) =h i~ δθi i t can be expressed as modes δθ k (t) andδ θ k (t) with wavenumber k. Thepositive and negative frequency part of these excitations are defined by δθ ± k ³δθ (t) =1 k (t) ± iδ 2 θ k (t)/ω k, δ θ ± k (t) = 1 ³ δ 2 θ k (t) iω k δθ k (t). Norm distribution: N i ± * i ± ± ± ± * k = δθ k δθ k δθ k δθ k

24 First mechanism: mode conversion (Unruh 96) When the Solutions to ω 0 = vk ± D() k lie within Brillouin zone: Lab -2 p.f n.f Positive frequency initial mode p.f comoving Late time (final) negative frequency pulse is depicted in Green. Initial negative frequency pulse in light blue (dashdotted), and positive frequency pulse in red (dashed).

25 In the commoving frame Wave remains a right mover Unruh s process

26 Second mechanism: Block-like oscillations (Jacobson 98) Soltions to ω = vk ± D k lie outside the Brillouin zone: 0 ( ) p.f p.f -1.0 Smaller velocity

27 In the commoving frame Scattering from right to left mover

28 Testing Hawking s hypothesis We compare between the occupation number of the outcoming Wave packet under the thermal Hypothesis And the the Bogoliubov coefficient: β T Nk ( final) = = k e ωk k T B 1 k N + k ( initial) For different late timewaves: 2 k 2πn ( ) 40π n δθ ( 0) = ke, n = 1,,20, k Which taken to be localized at flat space.

29 Hawking temperature The Black Hole horizon is located at c(θ H )=v(θ H ). The temperature T H of Hawking radiation is given by the expression (c depends on θ) kt B H 1 d 1 d = c v = v c 4πvdθ 2π dθ 2 ( 2 ) H ( ) H. In the case of a Coulomb chain with nearest neighbor interactions only, where the sound velocity of an homogeneous system c=\sqrt{2e^2n}, it can be evaluated in the local density approximation as kt B H g g = 1 3 ( ( θ )) 1 4 π g ( g ( θ) ) θ = θh

30 Numerical results 1. If only nearest neighbor Coulomb interactions are considered, the relative difference between these quantities is lower than ²<0.01 in our calculations with up to N=1000 ions. 2. For the long range Coulomb interactions it is of the order of ²<0.14. Here the definition of a global Hawking temperature is difficult because of the nonlinear dispersion relation at low wavenumbers. Possibly also due to the non-conformal scattering. 3. We have checked, that anharmonic effects do not significantly alter these results for oscillation amplitudes comparable to h θ 2 i at the Hawking temperature. Therefore qualitatively Hawking radiation (mode mixing) seems to persists even in a fundamentally discrete system with long range interactions and a logarithmically diverging group velocity at low wavenumbers

31 Experimental Parameters The ion velocity must lie in the same order of magnitude as the phonon velocity in the proposed experimental setup. For N=1000 singly charged 7 Li ions with an average spacing of L/N=10µm the rotation frequency of the ions must be ω ion =6.3kHz. The Hawking temperature in this system is k B T H /~=9.8ω ion =62kHz. These parameters and such a temperature can be realized experimentally.

32 Creation of a black hole Experimental sequence to measure evidence for Hawking radiation on ion rings: Begin with a thermal state of the excitations around homogeneously spaced subsonic rotating ions. Create in a short time a supersonic region to avoid white hole and finite size effects. We change the velocity profile parameter in an exponentially smooth way. In our case we used 0.01 < T creation T < 0.1 So we have a time of order up to 0.5 T to observe the radiation.

33 in Stability Deviation of the ions position relative to equilibrium as function Of time. BH Creation time is here ~0.01.

34 Correlations We analyze the correlations obtained from the covariance matrix Γ Local creation operators: 1 a = ( f δθ if δ p 2 1/4 1/4 i ii i ii i C Where we use Wick s theorem: ij () t = ( nˆ nˆ )( nˆ nˆ ) i i j i nˆ nˆ ha i a j a ia j i = ha i a iiha j a ji + ha i a j iha ia j i + ha i a jiha j a ii i j ) Note that the interpretation is quite different compared to the BEC case!

35 Density-density correlations t=0,0.1,0.2,0.3,0.4t R=2.25 bh correlations

36 On the left Non-normalized correlations. t=0.4t Possible breakdown of the harmonic approximation.

37 Testing kinematics by modifying the charge e 2 The Black hole is not created. Different in/out relative velocities

38 Summary (a) We suggest another avenue towards realization of an analogue BH with cold ions. (b) This fully discrete Physical analogue model agrees with previously suggested theoretical models. (c) It appears to be accessible in today's experiments with ions. (d) Various other tests can be easily done in this model.

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