Multi-Orbital Quantum Phase Diffusion
The collapses and revivals of a coherent matter wave field of interacting particles can serve as a sensitive interferometric probe of the interactions and the number statistics of the underlying quantum field. Here we show how the ability to observe long time traces of collapse and revival dynamics of a Bose-Einstein condensate loaded into a three-dimensional (3D) optical lattice allowed us to directly reveal the atom number statistics and the presence of effective coherent multi-particle interactions in a lattice. The multi-particle interactions are generated via virtual transitions to higher lattice orbitals and can find use for simulations of effective field theories with ultracold atoms in optical lattices Johnson et al. (2009). We measured their absolute strengths up to the case of six-particle interactions and compare our findings with theory.
Coherent quantum states represent the most robust and stable field solutions in physics Glauber (1963). Characterized by a single amplitude and phase, they have found widespread use in the quantum description of classical coherent fields, ranging from laser light to coherent matter waves in superconductors, superfluids or atomic Bose-Einstein condensates. Whenever interactions between the underlying particles are present, or - more generally - whenever the phase evolution of the number states (Fock states) that form the coherent state is nonlinear in particle number, coherent states can undergo an intriguing sequence of collapses and revivals. In such a sequence, the quantum state first evolves into a highly correlated and entangled state where at the time of the collapse the classical field vanishes, however, at a later time the entanglement is unraveled again and the original classical field is ideally recreated.
Remarkable examples of such collapses and revivals have been observed for a coherent light field interacting with a single atom in Cavity Quantum Electrodynamics Brune et al. (1996); Haroche and Raimond (2006), for a classical oscillation of a single ion held in a trap Meekhof et al. (1996); Poschinger et al. (2009) or for a matter wave field of a Bose-Einstein condensate Greiner et al. (2002); Anderlini et al. (2006); Sebby-Strabley et al. (2007). In the latter case, where the dynamics are induced by non-linear two-particle interactions between the atoms, the underlying process of quantum phase diffusion Yurke and Stoler (1986); Wright et al. (1996, 1997); Walls et al. (1997); Imamoglu et al. (1997); Castin and Dalibard (1997) has been proposed to be useful for the creation of highly correlated and entangled quantum states, such as large Schrödinger cat or spin-squeezed states Sorensen et al. (2001); Micheli et al. (2003); Pezzé and Smerzi (2009), the latter of which have been demonstrated recently Estève et al. (2008). In all these cases, it is typically assumed that the atoms occupy a single spatial orbital of the system. Atom-atom interactions can however promote particles to higher lying orbitals. The admixture of these excited orbitals modifies the shape of the spatial wavefunction and gives rise to renormalized interaction energies depending on the atom number, in close analogy to the so-called configuration interaction in quantum chemistry Szabo and Ostlund (1989). An instructive way to envisage the physical processes is provided by effective field theory: Virtual transitions of atoms to excited orbitals generate effective multi-particle interactions as higher order corrections to the single-orbital two-particle interaction Johnson et al. (2009). Previous experiments have only allowed the observation of few cycles of quantum phase diffusion dynamics Greiner et al. (2002); Anderlini et al. (2006); Sebby-Strabley et al. (2007) and therefore could not reveal the striking signatures and consequences of multi-orbital effects, which are expected to lead to a multitude of novel many-body quantum phases Alon et al. (2005); Zhao and Liu (2008). Here, we have been able to observe more than 40 collapses and revivals in long time traces of the quantum evolution of coherent matter wave fields trapped in a 3D optical lattice potential. Multi-orbital effects give rise to a discrete set of distinct two-particle interaction energies for different Fock states constituting a coherent field. These in turn result in characteristic beat signals in the collapse and revival time traces, from which we infer the occupation of Fock states, measure their energies with high precision and determine the strength of the effective multi-particle interactions. Recently, multi-particle interactions have been identified via loss resonances for three- and four-body recombination Kraemer et al. (2006); Knoop et al. (2009); Zaccanti et al. (2009), however coherent multi-particle interactions, which we demonstrate here in the form of effective interactions, have so far defied observation.
Let us consider a single site of a deep optical lattice filled with atoms occupying the lowest single-particle state of the system . Assuming weak interactions and excluding multi-orbital effects, collisions between the atoms lead to an energy shift of the single-orbital Fock state energy given by representing the onsite interaction energy of the Bose-Hubbard model. Here denotes the two-particle interaction energy, determined by the onsite wavefunction , the -wave scattering length and the mass of an atom . Within the restriction to the lowest vibrational state of the system, is independent of the filling at the lattice site.
Inducing virtual transitions to higher vibrational states, interactions modify the shape of the ground state wavefunction (Fig. 1B) and the two-particle interaction energy itself becomes atom number dependent Lühmann et al. (2008); Johnson et al. (2009); Lutchyn et al. (2009); Alon et al. (2005); Li et al. (2006); Hazzard and Mueller (2009). In this case, the multi-orbital Fock state energies in a single lattice well can be approximated by the expansion
representing the eigenenergies of the corresponding effective Hamiltonian Johnson et al. (2009). Here, coherent multi-particle interactions become explicitly visible with the characteristic strength of the -particle interactions being denoted by .
An efficient way to experimentally probe the eigenenergies of a Hamiltonian, is to monitor the non-equilibrium dynamics of a quantum state prepared in a superposition of different eigenstates. In our case, the state is a superposition of different atomic Fock states and can be expressed in the general form . Experimentally, a 3D array of such states is created by loading a BEC into a shallow 3D lattice potential (Fig. 1A). The time evolution of the ensemble can be probed by analyzing the visibility of the atomic interference pattern as observed after rapid switch-off of the lattice potential and subsequent time-of-flight expansion Gerbier et al. (2005). For an array of identical states , the visibility of the interference pattern relates monotonically to , where denotes the annihilation operator at a lattice site. Hence, the dynamical evolution of the matter wave field, the quantum phase diffusion of , is given by:
To illustrate the essential dynamics of Eq. (2), we assume the initial state to be a coherent state with , where denotes the complex field amplitude with the mean atom number and initial phase . In a single-orbital model with eigenenergies , periodic collapses and revivals of the visibility are expected, which can analytically be expressed by featuring a single frequency and its higher harmonics (Fig. 1C (gray solid line)) Imamoglu et al. (1997); Castin and Dalibard (1997).
Multi-orbital effects, however, reach beyond the picture of monochromatic matter wave dynamics: the time-evolution of contains multiple frequency components since Fock state energies are no longer integer multiples of , as shown in Fig. 1C (blue solid line). Assuming repulsive interactions, we expect frequencies of order to be given by as well as higher order frequency components , with being a positive integer.
The detection of multi-orbital quantum phase diffusion over long times results in a high spectral resolution, allowing for a precise measurement of individual Fock state energies and multi-particle interaction strengths. In addition, Eq. (2) implies that the spectral weight of each contributing frequency is directly connected to the coefficients of the initial state . Hence, the statistical contribution of single Fock states to the global superfluid quantum many-body state can be obtained complementary to previous work Gerbier et al. (2006); Campbell et al. (2006); Cheinet et al. (2008).
Our experiments began with an atomic Bose-Einstein condensate of repulsively interacting Rb atoms in the hyperfine state, with variable atom numbers between and . The atoms were initially held in a pancake-shaped crossed optical dipole trap at a wavelength of nm with trap frequencies Hz in the direction of gravity and Hz in the orthogonal plane. Subsequently a 3D optical lattice ( nm) of simple cubic type was adiabatically ramped up to lattice depths between , where denotes the recoil energy. For these lattice depths the many-body ground state of the system is a superfluid and we expect the states on a lattice site to range from coherent states for shallow lattice depths to highly number squeezed states for deeper lattices Greiner et al. (2002); Gerbier et al. (2006); Estève et al. (2008); Cheinet et al. (2008).
A sudden increase of the lattice depth from to ranging between strongly suppressed the tunnel coupling and essentially froze out the atom number distribution on each site. In this regime, the time evolution of each site is governed by the Fock state energies of Eq. (1) and the quantum phase diffusion process started. The jump to was carried out within which is slow enough to avoid populating higher vibrational states, but fast compared to tunneling within the first band. Previous experiments were limited to follow quantum phase diffusion only for short traces with a maximum of 4-5 collapse and revival cycles Greiner et al. (2002); Anderlini et al. (2006); Sebby-Strabley et al. (2007) since a global harmonic confinement led to rapid relative dephasing of lattice sites. Here, the different detunings of the optical lattice (blue-detuned) and the dipole trap (red-detuned) with respect to the atomic resonance ( nm) allowed us to change the underlying harmonic confinement during the experimental sequence (Fig. 2C). Along with the jump to we instantaneously reduced the dipole trap in order to cancel harmonic confinement in the horizontal plane, boosting the coherence time of quantum phase diffusion. Under such conditions, the atomic sample stayed trapped due to the presence of the deep optical lattice . After letting the system evolve for a hold time , all trapping potentials were switched-off simultaneously and an absorption image of the matter wave interference pattern was recorded after 10 ms time-of-flight. The visibility of the interference pattern was evaluated, being a measure for the ensemble average of Gerbier et al. (2005).
A typical time trace () of quantum phase diffusion is shown in Fig. 2A displaying about 40 revivals. On top of a fast series of collapses and revivals, we observe a slower modulation of the envelope indicating a beat between at least two different energies in the system. The spectral content of the time trace has been quantitatively analyzed using a numerical Fourier transform of the data (Fig. 2B). We clearly identify three distinct frequency components of order present in the time trace, the smallest one originating from sites occupied by up to four atoms. It is remarkable that our technique can reveal even very small Fock state amplitudes due to a heterodyne effect with other Fock states and of typically larger amplitudes and (Eq. 2). The residual damping on the time trace most likely results from residual harmonic confinement (along the -direction), residual tunneling Fischer and Schützhold (2008) via higher bands and the finite extension of the atomic ensemble, sampling a slightly inhomogeneous distribution of lattice depths due to the Gaussian shape of the lattice laser beams.
In order to map out the dependence of the multi-orbital Fock state energies on lattice depth, we recorded several long collapse and revival traces with between (Fig. 3). With a relative statistical uncertainty on the level of few percent, a Fourier analysis reveals both a set of frequencies of order and , monotonically increasing with lattice depth. The measured two-particle Fock state energy is about % lower than predicted by a simple single-orbital Bose-Hubbard model, which strongly indicates the relevance of multi-orbital effects in optical lattices (Fig. 3A).
We compare our experimental data to an ab-initio exact diagonalization of a single-site multi-orbital system. For this system a contact interaction with an -wave scattering length of Klausen et al. (2001) was assumed, where denotes the Bohr radius. We find excellent agreement, particularly for deep lattices ( ), when the diagonalization is performed on a Hilbert-space with single-particle orbitals, corresponding to the four lowest lattice bands in three dimensions (Fig. 3A, black solid lines). For lower lattice depths ( ) the results of a calculation including onsite orbitals appear even more suitable (Fig. 3A, gray dashed lines). The remarkable congruence between theory and experiment suggests, that the influence and occupation of even higher lying orbitals is negligible.
The strength of the effective multi-particle interactions has been iteratively derived from the measured multi-orbital energies according to , and (Fig. 3B). The resulting effective three-particle interaction is attractive and the measured values agree well with the results obtained from exact diagonalization, indicating that direct three-particle interactions are negligible for our experimental parameters. The measured effective four-particle interaction strengths on a scale as low as Hz are also in good agreement with theory, taking the experimental uncertainty into account. Our data shows, that the expansion provided by Eq. 1 quickly converges, offering the possibility to efficiently incorporate the effects of multi-orbital physics in refined effective single-band lattice Hamiltonians.
In addition to its capability to measure coherent interatomic interaction strengths, multi-orbital quantum phase diffusion can be used to reveal the number statistics of many-body quantum states: The spectral weight of the detected frequencies carries information on the Fock state amplitudes (Eq. 2). To demonstrate this, we adiabatically prepared 3D arrays of coherent ( ) to highly number-squeezed states ( ) close to the Mott-insulator transition around , and employed quantum phase diffusion as a detection sequence (see Fig. 4A). From to the time traces evolve from seemingly irregular oscillations into a clear beat signal since less frequencies contribute, as can be observed in the corresponding Fourier spectra (Fig. 4B). This narrowing of the spectrum reflects a decreasing variance of the atom number distribution caused both by number squeezing and the reduction of the average onsite density due to increased interactions when the lattice depth is raised.
In conclusion, we have demonstrated how interaction induced virtual transitions to higher lattice orbitals lead to effective coherent multi-particle interactions Johnson et al. (2009). We have determined their strengths by monitoring long time traces of the collapse and revival dynamics of the matter wave field of a BEC. Our measurements explicitly demonstrate the coherence of the multi-particle interactions and the number state resolving capabilities of multi-orbital quantum phase diffusion. Multi-orbital quantum phase diffusion can serve as a precise experimental technique to determine renormalized interaction energies in lattice based quantum gases, which have been found to play a crucial role in more complex systems such as Bose-Fermi mixtures Lühmann et al. (2008); Best et al. (2009); Lutchyn et al. (2009). Our results also show that multi-orbital effects will play an important role in experiments working towards interaction induced dynamical creation of spin-squeezing or Schrödinger cat states with ultracold atoms. Furthermore, we envisage that the coherent multi-particle interactions demonstrated here could be employed to simulate effective field theories Johnson et al. (2009) and also help in realizing novel strongly correlated many-body quantum phases Alon et al. (2005); Hazzard and Mueller (2009), e.g. with topological order Paredes et al. (2007) or exotic ground state properties Capogrosso-Sansone et al. (2009).
This work was supported by the DFG, the EU (IP SCALA), EuroQUAM (LH), DARPA (OLE program), AFOSR, MATCOR (SW) and the Gutenberg Akademie (SW)
- Johnson et al. (2009) P. R. Johnson, E. Tiesinga, J. V. Porto, and C. J. Williams, N. J. Phys. 11, 093022 (2009).
- Glauber (1963) R. J. Glauber, Phys. Rev. 131, 2766 (1963).
- Brune et al. (1996) M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 76, 1800 (1996).
- Haroche and Raimond (2006) S. Haroche and J.-M. Raimond, Exploring the Quantum, Oxford Graduate Texts (Oxford University Press, 2006).
- Meekhof et al. (1996) D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).
- Poschinger et al. (2009) U. G. Poschinger, G. Huber, F. Ziesel, M. Deiss, M. Hettrich, S. A. Schulz, K. Singer, F. Schmidt-Kaler, G. Poulsen, M. Drewsen, et al., arXiv:0902.2826v2 (2009).
- Greiner et al. (2002) M. Greiner, M. O. Mandel, T. Hänsch, and I. Bloch, Nature 419, 51 (2002).
- Anderlini et al. (2006) M. Anderlini, J. Sebby-Strabley, J. Kruse, J. V. Porto, and W. D. Phillips, J. Phys. B 39, S199 (2006).
- Sebby-Strabley et al. (2007) J. Sebby-Strabley, B. Brown, M. Anderlini, P. Lee, P. Johnson, W. Phillips, and J. Porto, Phys. Rev. Lett. 98, 200405 (2007).
- Yurke and Stoler (1986) B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
- Wright et al. (1996) E. M. Wright, D. F. Walls, and J. C. Garrison, Phys. Rev. Lett. 77, 2158 (1996).
- Wright et al. (1997) E. M. Wright, T. Wong, M. J. Collett, S. M. Tan, and D. F. Walls, Phys. Rev. A 56, 591 (1997).
- Walls et al. (1997) D. F. Walls, M. J. Collett, T. Wong, S. M. Tan, and E. M. Wright, Phil. Trans. R. Soc. A 355, 2393 (1997).
- Imamoglu et al. (1997) A. Imamoglu, M. Lewenstein, and L. You, Phys. Rev. Lett. 78, 2511 (1997).
- Castin and Dalibard (1997) Y. Castin and J. Dalibard, Phys. Rev. A 55, 4330 (1997).
- Sorensen et al. (2001) A. Sorensen, L. M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001).
- Micheli et al. (2003) A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. A 67, 013607 (2003).
- Pezzé and Smerzi (2009) L. Pezzé and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009).
- Estève et al. (2008) J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature 455, 1216 (2008).
- Szabo and Ostlund (1989) A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (McGraw-Hill, 1989).
- Alon et al. (2005) O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Rev. Lett. 95, 030405 (2005).
- Zhao and Liu (2008) E. Zhao and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008).
- Kraemer et al. (2006) T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H. C. Nägerl, et al., Nature 440, 315 (2006).
- Knoop et al. (2009) S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Schobel, H. C. Nagerl, and R. Grimm, Nat. Phys. 5, 227 (2009).
- Zaccanti et al. (2009) M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. Muller, G. Roati, M. Inguscio, and G. Modugno, Nat. Phys. 5, 586 (2009).
- Lühmann et al. (2008) D.-S. Lühmann, K. Bongs, K. Sengstock, and D. Pfannkuche, Phys. Rev. Lett. 101, 050402 (2008).
- Lutchyn et al. (2009) R. M. Lutchyn, S. Tewari, and S. D. Sarma, Phys. Rev. A 79, 011606 (2009).
- Li et al. (2006) J. Li, Y. Yu, A. M. Dudarev, and Q. Niu, N. J. Phys. 8, 154 (2006).
- Hazzard and Mueller (2009) K. R. A. Hazzard and E. J. Mueller, arXiv:0902.4707v1 (2009).
- Gerbier et al. (2005) F. Gerbier, A. Widera, S. Fölling, O. Mandel, T. Gericke, and I. Bloch, Phys. Rev. Lett. 95, 050404 (2005).
- Gerbier et al. (2006) F. Gerbier, S. Fölling, A. Widera, O. Mandel, and I. Bloch, Phys. Rev. Lett 96, 090401 (2006).
- Campbell et al. (2006) G. Campbell, J. Mun, M. Boyd, P. Medley, A. E. Leanhardt, L. G. Marcassa, D. E. Pritchard, and W. Ketterle, Science 313, 5787 (2006).
- Cheinet et al. (2008) P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M. Moreno-Cardoner, S. Fölling, and I. Bloch, Phys. Rev. Lett. 101, 090404 (2008).
- Fischer and Schützhold (2008) U. R. Fischer and R. Schützhold, Phys. Rev. A 78, 061603 (2008).
- Klausen et al. (2001) N. N. Klausen, J. L. Bohn, and C. H. Greene, Phys. Rev. A 64, 053602 (2001).
- Best et al. (2009) T. Best, S. Will, U. Schneider, L. Hackermüller, D. van Oosten, I. Bloch, and D.-S. Lühmann, Phys. Rev. Lett. 102, 030408 (2009).
- Paredes et al. (2007) B. Paredes, T. Keilmann, and J. I. Cirac, Phys. Rev. A 75, 053611 (2007).
- Capogrosso-Sansone et al. (2009) B. Capogrosso-Sansone, S. Wessel, H. P. Büchler, P. Zoller, and G. Pupillo, Phys. Rev. B 79, 020503 (2009).
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to S. Will (email: email@example.com)