Probing the Optical Conductivity of Harmonically-confined Quantum Gases!

Transcription

1 Probing the Optical Conductivity of Harmonically-confined Quantum Gases! Eugene Zaremba Queen s University, Kingston, Ontario, Canada Financial support from NSERC Work done in collaboration with Ed Taylor and Zhigang Wu

2 The Art of Poking and Prodding! parametric perturbations V trap (x, t) = 2 m! 0(t) 2 x 2! 0 (t) =! 0 +! 0 (t),! 0 (t) =! 0 +! 0 sin t modulation spectroscopy V trap (x)+v opt (x, t) V opt (x, t) =V 0 (t)sinkx, V opt (x, t) =V 0 sin k(x x 0 (t)) effective dynamics V trap (x)+v ext (x) (x)! (x x 0 ) ) V ext (x x 0 cos! 0 t) oscillating harmonic potential optical conductivity V trap (x, t) = 2 m!2 0(x x 0 sin!t)

3 Experiment on Disorder-induced Damping! oscillations of the centre of mass are induced by suddenly shifting the harmonic confining potential a disorder potential created by a laser speckle pattern leads to the decay of the centre of mass motion Y.P. Chen et al., Phys. Rev. A77, (2008)

4 Equation of Motion in Presence of Disorder! the Hamiltonian of the system is Ĥ = NX i= apple ˆp 2 i 2m + V trap(ˆr i ) + ˆV int + NX V dis (ˆr i ) Ĥ0 + ˆV dis i= for an arbitrary dynamical state, the c.m. coordinate Z(t) = (t) ˆR z (t) satisfies d 2 Z dt zz = F M where F (t) = (t) b F z (t) bf z = NX i= V dis (ˆr i ) ẑ i evaluation of the force requires knowledge of the dynamical state

5 Extended Harmonic Potential Theorem! 0 0 in (a), the condensate is initially displaced relative to the centre of the harmonic trap and oscillates in the presence of a static disorder potential in (b), the condensate is initially at rest at the centre of the harmonic trap but the disorder potential is displaced and is made to oscillate at the trap frequency the force is exactly the same in these two physically distinct scenarios D. Dries et al., Phys. Rev. A82, (200)

6 Linear Response! the utility of this latter point of view is that the force can be calculated using conventional linear response theory if the perturbation is weak to second order in the disorder potential, we have F (t) = i ~ where = Z t 0 Z t 0 dt 0 0 [ e F z,i (t), e V dis,i (x(t 0 ),t 0 )] 0 Z dt 0 dkz 2 R(k z)k z e ik z[z(t) z(t 0 )] (k z,k z ; t t 0 ) (k z,k z ; t t 0 )= R dr R dr 0 e ik z(z z 0 ) (r, r 0 ; t t 0 ) (r, r 0 ; t t 0 )= i ~ (t t0 ) 0 [ˆn I (r,t), ˆn I (r 0,t 0 )] 0 the disorder-averaged speckle pattern gives R(k z )= p V 2 dis e 4 2 k 2 z

7 Density Response Function! within the Bogoliubov approximation, the density response function is determined by the collective excitations of the system (r, r 0 ; t t 0 )= i ~ (t t0 ) P i n o n i (r) n i (r0 )e i i(t t 0 ) c.c. the Bogoliubov excitations for an arbitrary anisotropic trap must be determined numerically a simpler approach is to make use of the cylindrical local density approximation, a variant of the more commonly used bulk local density approximation (k z,k z ; ) ' R dz cyl (k z, ; (z)) in this approximation, the condensate responds to a perturbation as if it were locally part of an infinitely long cylindrical condensate with density per unit length ν(z)

8 Damping of the Centre of Mass Oscillation! Y.P. Chen et al., Phys. Rev. A77, (2008) the damping of the mode increases with increasing strength of the disorder potential Z(t) =z 0 e bt cos zt theoretical estimate of damping d 2 Z dt 2 + zz 2 = F M Z l Z(T l ) Z(T l )= M z Z Tl T l dt sin z tf (t) b ' Z z 0 T ' Z z 0 T

9 Results*! b z = 2 Mv 2 0 ~ Z Rz Z dk dz R z 2 R(k) X j X j 2 (k) njn(z 2 0 k) ( j (k) n z ) n= Experimental Comparison initial displacement z µm v 0 /c disorder strength damping V 2 dis /µ2 ' (b/ z ) th ' (b/ z ) exp ' the damping exhibits a resonant peak for a velocity equal to the sound speed at the centre of the trap the results for the bulk LDA differ quantitatively from the cylindrical LDA *Z. Wu and E. Zaremba, Phys. Rev. Lett. 06, 6530 (20)

10 Other Data*! we do not find good agreement for this set of data we have no explanation for this for comparable experimental parameters, the Chen et al. experiment seems to give different results from the Dries et al. experiment our theory can only account for the damping at early times immediately after excitation since the cloud heats up significantly during the course of the evolution *D. Dries et al., Phys. Rev. A82, (200)

11 Shaking Potentials*! we consider a harmonically-trapped system in the presence of a time-dependent external potential Ĥ 0 = Ĥ(t) =Ĥ0 + NX i= NX V ext (ˆr i! i= ˆp 2 i 2m + V trap(ˆr i ) for small displacements r 0 (t), with Ĥ 0 (t) = Ĥ(t) =Ĥ + Ĥ0 (t) Ĥ = Ĥ0 + NX V ext (ˆr i ) i= NX rv ext (ˆr i ) r 0 (t) = i= + X i<j r 0 (t)) v(ˆr i ˆr j ) the perturbation is seen to couple to the particle density Z drrv ext (r) r 0 (t)ˆn(r). *Z. Wu and E. Zaremba, Ann. Phys. 342, 24 (204)

12 Perturbative Analysis! the perturbation can be rewritten as Ĥ 0 (t) = X r 0µ (t) m NX i~ [Ĵµ, V ext (ˆr i )] = X µ i= µ r 0µ (t) hâµ + Â2µ i  µ = m i~ [Ĵµ, Ĥ],  2µ = Nm! µ 2 ˆR µ Ĵ µ = m ˆP µ = NX ˆp iµ, ˆRµ = NX m N i= the energy absorption rate is given quite generally by dẽ dt = h (t) dĥ0 (t) dt (t)i in linear response one finds dẽ dt = X iµ ṙ 0µ (t)h 0 Âiµ 0i X iµ,j ṙ 0µ (t) Z where we have defined the retarded response functions i= ˆr iµ iµ,j (t t 0 ) i ~ (t t0 )h 0 [Âiµ(t), Âj (t 0 )] 0i dt 0 iµ.j (t t 0 )r 0 (t 0 )

13 Perturbative Analysis, cont d! for a monochromatic displacement of the form r 0µ (t) = 2 (r 0µe i!t + r 0µe i!t ) the time-averaged energy absorption rate is found to be dẽ dt = m2! 3 2 X r0µr 0 µ the energy absorption rate is directly related to the optical conductivity defined as! 2 µ! 2! 2! 2 Im µ (!) where Im µ (!) is the imaginary part of the Fourier transform of the current-current response function µ (t t 0 ) i ~ (t t0 )h 0 [Ĵµ(t), Ĵ (t 0 )] 0i Re µ (!) =! Im µ (!)

14 Discussion! for a displacement of the form dẽ dt = 2 m2! 3 z 2 0 r 0 (t) =z 0 ẑ sin!t! 2 z! 2 2 Im zz(!) we have the current response includes the full effect of the external potential and the harmonic trapping potential information about the possible excitations in the system is contained in the frequency dependence of the current response function in the limit! z! 0 one recovers the result of Tokuno and Giamarchi (PRL 06, (20)). It should be noted however that their derivation is not quite right. one sees that the energy absorption rate vanishes when! =! z. This is a consequence of the extended harmonic potential theorem (Wu and Zaremba, Ann. Phys. 342, 24 (204)) and is not limited to the perturbative analysis. the low-frequency spectral density is enhanced in the presence of the trap

15 MIT Experiment on Superfluidity [PRL 85, 2228 (2000)]! a gaussian potential is rastered periodically through an elongated cylindrical condensate the induced density exhibits an asymmetry when dissipation sets in the dissipation rate is small when the rastering frequency is close to the axial trapping frequency

16 Measuring the Optical Conductivity*! we consider an oscillating trapping potential of the form with V trap (r,t) = X µ = V trap (r) 2 m!2 µ[x µ x 0µ (t)] 2 X F µ (t)x µ + X µ µ F µ (t) =m! 2 µx 0µ (t) 2 m!2 µx 2 0µ(t) the displacement of the trap leads to a time-dependent homogeneous force acting on the system and results in a perturbation that couples to the centre-of-mass coordinate H 0 (t) = X µ NF µ (t) ˆR µ this coupling can be treated perturbatively if F µ (t) is small *Z. Wu, E. Taylor and E. Zaremba, EPL 0, (205)

17 Perturbation Analysis! the centre of mass coordinate is the physical quantity of interest within linear response theory, one finds Z h ˆR µ (t)i = Fourier transforming we have the total current operator is dt 0 R µ (t t 0 )NF (t 0 ) where the centre of mass response function is R µ (t t 0 )= i ~ (t t0 )h[ ˆR µ (t), ˆR (t 0 )]i R µ (!) = Rµ (!)NF (!) Ĵ µ (t) =N d ˆR µ (t) dt and the total induced current is given by J µ (!) = µ (!)F (!) where the global conductivity is µ (!) = i!n 2 Rµ (!)

18 Observables! we now choose the force to be in a specific direction conductivity tensor is then given by ; the furthermore, if the trap displacement is harmonic at the frequency! 0, we have and (!) = i!n F (!) R (!) R (t) =A (! 0 ) cos[! 0 t (! 0 )] (! 0 )= i! 0N F (! 0 ) A (! 0 )e i (! 0 ) = thus, a measurement of the amplitude and phase of the centre of mass displacement as a function of the oscillation frequency ω 0 determines the optical conductivity the optical conductivity is also related to the total current correlations via with Jµ (!) =F apple N m µ µ (!) = i! apple i ~ ( )h[ĵµ( ), Ĵ (0)]i = F [ µ ( )] Jµ (!)

19 Application: D Bose-Hubbard Model! we consider N atoms in a D optical lattice with harmonic confinement; in the tight-binding limit, the Hamiltonian is Ĥ BH = X i iˆn i t X i and the current operator is â i âi+ +â i+âi + 2 U X i ˆn i (ˆn i ) in this model, the centre-of-mass coordinate is given by ˆR = a N Ĵ N i~ [ ˆR, ĤBH] = the optical conductivity is (!) = i! X iˆn i i ta i~ apple a 2 X j â jâj â j âj ~ 2 h ˆT i J(!)

20 Calculation of the Current Response! the imaginary part of the current response function is given by Im J(!) = X h 0 Ĵ i 2 [ (~! E 0 ) (~! + E 0 )] and the real part is obtained from a KK transform the eigenstates and eigenenergies are obtained deep in the Mottinsulator regime t/u ; here, the relevant excitations are single particle-hole excitations we consider a lattice with M sites containing N particles; the ground state is 0i = Y i2n â i 0i and the set of excited states retained are the particle-hole states phi =â pâ h 0i, h 2 N,p 2 M the B-H Hamiltonian is diagonalized using the above truncated Hilbert space

21 Numerical Results! 4 x 0 4 ˆ α 2 π Φ0 J Φ 3 E α0/u excitation energies vs. state index 0.5 E α0/u.5 current matrix elements vs. energy.5 the approximations yield an optical conductivity which accurately satisfies the exact sum rule Σ(ω) ω/u Z.5 the real and imaginary parts of the optical conductivity P. d! Re (!) = a2 ht i ~2 F. Maldague, PRB 6, 2437 (977)

22 Behaviour of the Centre of Mass! R(t) =A(!) cos[!t (!)] (!) = 4 3 i!n (!) A(!)ei F (!) A(ω) 3 φ(ω) ω/u amplitude vs. frequency ω/u phase vs. frequency.5 Σ(ω) ω/u

23 Summary! we have used the Extended Harmonic Potential Theorem to analyze the disorder-induced damping of the centre-of-mass motion of a Bose condensate we have analyzed the energy absorption rate for the case of a shaking external potential in the presence of harmonic confinement; the energy absorption rate is related to the optical conductivity and vanishes when ω = ω trap as a consequence of the extended harmonic potential theorem a complementary excitation scheme is to shake the confining harmonic potential keeping other external potentials fixed; this directly yields the optical conductivity one advantage of this latter scheme is that the optical conductivity can be determined experimentally by simply measuring the position and phase of the center of mass