We can then linearize the Heisenberg equation for in the small quantity obtaining a set of linear coupled equations for and :

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1 Wednesday, April 23, :37 PM Excitations in a Bose condensate So far: basic understanding of the ground state wavefunction for a Bose-Einstein condensate; We need to know: elementary excitations in a BEC. The Bogoliubov approximation As we did before: to go from classical light to quantum we quantized the electric field E. For matter waves, we will replace the field operator with its average value and some small fluctuations about that average We can then linearize the Heisenberg equation for in the small quantity obtaining a set of linear coupled equations for and : and similar for the hermitian conjugate. Since these coupled equations are linear, we can solve them by looking for normal mode solutions. We will define operators j and j which annihilate and create bosonic excitations in the normal mode with frequency ; since the excitations are bosons, their creation and annihilation operators obey Bose statistics [ k, j ]= jk. We can expand in terms of the normal modes and the field operator will be properly normalized if New Section 1 Page 1

2 The coefficients for each normal mode are independent, so we obtain a set of equations for each mode j. Equating terms e i jt and e -i jt, we find where = e iμt and a real number. These time independent equations give the eigenfrequencies j and eigenmodes {u j, v j } of excitations in the condensate, and are known as the Bogoliubov equations. Example: homogeneous potential The Bogoliubov equations are particularly transparent for a condensate in a uniform potential V (x) = 0. The Thomas-Fermi approximation gives where n 0 is the condensate density and μ = n 0 U 0 is the chemical potential. Since there is no explicit dependence on position anywhere in the problem, the eigenmodes are most easily obtained in the Fourier domain The Bogoliubov equations become To find the eigenvalues and eigenmodes associated with excitations in the homogenous condensate, we need only solve this matrix equation. The frequencies q follow from setting the determinant of the matrix to zero, yielding the dispersion relation New Section 1 Page 2

3 Note: the dispersion relation is qualitatively different from that of a free particle. In the large-wavenumber and small-wavenumber regimes, the frequency is which correspond to particle-like and sound-like excitations respectively. The sound-like spectrum has an associated sound velocity and the cross-over between the two regimes occurs near a critical wavevector Physically, the long-wavelength excitations correspond to collective excitations of the condensate, and are responsible for the phenomenon of superfluidity. Once the wavelength approaches the inverse healing length, the condensate ceases to behave as a continuum, and singleparticle-like excitations dominate. New Section 1 Page 3

4 Probing the excitation spectrum Experiments have measured the excitation spectrum in a gaseous Bose condensate using a Bragg scattering technique. Two lasers with frequencies 1, 2 and wavevectors k 1 and k 2 interact with the condensate transferring momentum q = k1 k2 to excitations within the cloud. When the energy difference is resonant with the energy of the excitation the lasers create a collective excitation. By measuring the spectrum of such resonances, one can determine the sound velocity in the condensate. Superfluidity of repulsive atoms Superfluidity requires that an object can move slowly through the fluid without experiencing any drag. Since drag is the macroscopic manifestation of excitations in a medium, we will see that the excitation spectrum of a repulsive Bose condensate implies superfluidity. In particular, consider an object moving at velocity v through a Bose condensate. It will only create an excitation if it can resonantly impart its momentum and energy to an elementary excitation: Kinematics require that, so the object can only create an excitation if it is moving faster than some critical velocity This represents the so-called Landau criterion for superfluidity, which states that objects moving slower than the speed of sound do not produce excitations New Section 1 Page 4

5 Potential flow and quantized circulation A BEC moving with velocity v can be described as From this it follows that the flow velocity is: This relationship has important consequences on the motion of a BEC. Form it follows that That is, the velocity field is irrotational, unless the phase of the order parameter has a singularity. Possible circulation of the BEC is restricted. From the single value of the condensate wave function, it follows that around a close contour the change of the wave function must be a multiple of 2 p or with l an integer. For a simple example consider a pure circulation in a trap invariant under rotation about z. If is the distance from the trap axis, it follows that the azimutal velocity If l>0 the wave function must vanish on the axis of the trap, since otherwise the kinetic energy will diverge BEC with attractive interactions If the interatomic potential is attractive U 0 < 0, long-wavelength excitations have an imaginary frequency q. This implies that the condensate New Section 1 Page 5

6 is unstable! Experimentally, however, stable condensates with attractive interactions have been observed. The explanation for the stability lies in the finite number of atoms confined to the trap. If an atomic cloud is localized within a characteristic trap length a t, the minimum wave-vector allowed by the system is of order so the excitations in the system have a minimum energy When this minimum energy is real, the condensate will be stable despite the attractive interactions. In particular, this implies a maximum number of atoms in the cloud If condensate size exceeds N max, the attractive interactions will destabilize it and it will collapse. Observation of BEC Typical experiments probe the cold atom cloud using absorption imaging with near-resonant light. In certain cases, one could image the condensate in the trap, obtaining a spatial profile of the atom cloud. Alternately, by releasing the cold atom cloud from the trap and allowing it to expand for a time t, the momentum distribution of the atoms p is mapped onto the spatial distribution r = pt/m. Such expansion images provide a means for distinguishing a cold, thermal atom New Section 1 Page 6

7 cloud from a condensate. In a condensate, the momentum distribution follows from the size of the atom cloud ac and the uncertainty principle, resulting in a post-expansion spatial distribution In contrast, a thermal cloud of atoms will have a gaussian distribution of momenta in the trap leading to a gaussian spatial distribution where R T and p T are the distance and momentum scale for an atom cloud at temperature T. The two momentum distributions are qualitatively different. By fitting the observed image to a bimodal distribution corresponding to a gaussian thermal cloud and a parabolic condensate, one can accurately calculate the proportion of condensed atoms. New Section 1 Page 7