Manipulating atoms in an optical lattice: Fractional fermion number and its optical quantum measurement

We provide a detailed analysis of our previously proposed scheme [J. Ruostekoski, G. V. Dunne, and J. Javanainen, Phys. Rev. Lett. 88, 180401 (2002)] to engineer the profile of the hopping amplitudes for atomic gases in a one-dimensional optical lattice so that the particle number becomes fractional. We consider a constructed system of a dilute two-species gas of fermionic atoms where the two components are coupled via a coherent electromagnetic field with a topologically nontrivial phase profile. We show both analytically and numerically how the resulting atomic Hamiltonian in a prepared dimerized optical lattice with a defect in the pattern of alternating hopping amplitudes exhibits a fractional fermion number. In particular, in the low-energy limit we demonstrate the equivalence of the atomic Hamiltonian to a relativistic Dirac Hamiltonian describing fractionalization in quantum field theory. Expanding on our earlier argument [J. Javanainen and J. Ruostekoski, Phys. Rev. Lett. 91, 150404 (2003)] we show how the fractional eigenvalues of the particle number operator can be detected via light scattering. In particular, we show how scattering of far-off resonant light can convey information about the counting and spin statistics of the atoms in an optical lattice, including state-selective atom density profiles and atom number fluctuations. Optical detection could provide a truly quantum mechanical measurement of the particle number fractionalization in a dilute atomic gas.

Figure 1

(Color online) The energy diagram of the two-species Fermi-Dirac gas in an optical lattice. The atoms occupy two different internal levels and experience different periodic optical potentials shifted by $d$ (on the left-hand side). The coupling between the neighboring lattice sites, representing different spin components, is induced by two-photon Raman transitions, or by a superposition of the Raman transition and a one-photon microwave or rf transition (dotted line). The transition region is denoted by the dark shaded area representing the overlap region of the atomic wave functions. The energy difference between the two components in the overlap region is denoted by $\omega$. The two-photon transition is far-detuned by $\Delta$ from an intermediate atomic level (on the right-hand side), and in this illustration $\delta$ stands for the two-photon detuning.

Figure 2

(Color online) Dimerized optical lattice with spatially alternating hopping amplitude for atoms between adjacent lattice sites. The hopping is induced by electromagnetic field (as in Fig. 1) whose amplitude varies according to Eqs. (5, 8). Here $a_k$ and ${\mu }_k$ are assumed to be constant along the lattice. The coefficient in the front of $\mu$ is determined by the value of $\text{sin}\left(\pi x/d\right)$ at the middle between the lattice sites where the Wannier functions of the two lattice sites overlap.

Figure 3

(Color online) Defect in the pattern of alternating hopping amplitudes. The coefficient in the front of $\mu$ is determined by the product $\phi \left(x\right)\text{sin}\left(\pi x/d\right)$ at the middle between the lattice sites where the Wannier functions of the two lattice sites overlap. The soliton, or a kink, profile of $\phi \left(x\right)$ generates a defect in the hopping amplitude pattern, so that the same coupling matrix element appears 2 times. The defect is marked by a dashed (red) vertical line. Here $\phi \left(x\right)\sim x/\left|x\right|$.

Figure 4

(Color online) Examples of the field configurations that are not based on an optical hologram and that can generate the field amplitude in Eq. (19). Two different transition paths synthesize the superposition between $\mathcal{V}$ and $\phi \left(x\right)\text{sin}\left(\pi x/d\right)$. The field profile $\phi \left(x\right)\text{sin}\left(\pi x/d\right)$ is produced by two successive transitions, so that the effective Rabi frequency is the product of $\phi \left(x\right)$ and $\text{sin}\left(\pi x/d\right)$. On top, a far-detuned microwave or rf field, generating $\phi \left(x\right)$, is combined with an optical Raman two-photon transition via a far-detuned electronically excited intermediate level, where one of the transitions is driven by the standing wave $\text{sin}\left(\pi x/d\right)$. The effective Rabi frequency for the corresponding three-photon transition is proportional to $\propto \phi \left(x\right)\text{sin}\left(\pi x/d\right)$. On bottom, an optical Raman two-photon transition where one transition is driven by $\phi \left(x\right)$ and the other one by $\text{sin}\left(\pi x/d\right)$.

Figure 5

A scheme for a dimerized lattice based on the interplay between internal and center-of-mass energies of the atoms. (a) The energies of the relevant internal states ($\uparrow$ and $\downarrow$) of an atom at lattice sites $j=-2,\dots ,2$ are marked by horizontal bars, with resonant transitions between the states displayed by solid arrows. (b) After a constant energy difference between adjacent lattice sites is added to the atoms, for a given set of driving fields only one-half of the transitions can be on resonance. (c) Another set of laser frequencies is added, which render also the remaining transitions resonant (dashed arrows). A dimerized lattice has been prepared. Several transitions without a resonant final state are also indicated. (d) In a variation of the scheme in part (b), a constant energy difference is added between every state pair to the left-hand side of $j=0$, and the negative of the same energy difference between state pairs to the right-hand side of $j=0$. The same coupling as in part (c) now realize a dimerized lattice with a defect.

Figure 6

Coherent atom number fluctuations $\Delta N$ under a Gaussian (solid line) and alternating sign Gaussian (dashed line) envelopes, ${\alpha }_k^G$ and ${\alpha }_k^A$, and the corresponding incoherent atom number fluctuations ${\left(\Delta N\right)}_i$ (dotted line) as a function of the width of the Gaussian $w$. The lattice parameters are $N_s=1025$, $N_f=513$, and $\mu =-0.1a$.

Figure 7

(Color online) The nonuniform spatial profile for the hopping amplitudes. (a) Here $\pm$ denote the couplings $a\pm \mu$ in Eq. (8) where one of the signs of the hopping amplitudes has been changed. Introducing the sign switch generates two defects that are marked by the vertical lines. (b) Here $\pm$ denote the couplings $a\pm \mu$ in Eq. (44), with $A_k^{\left(p\right)}$ given by Eq. (45), where one of the signs in the middle has been switched. In this case the sign change introduces three defects (marked by vertical lines). The first defect represents the boundary between the $p=0$ and $p=2$ threefold degenerate configurations of Eq. (45); the second between $p=2$ and $p=1$; and the third between $p=1$ and $p=0$. Part (c) represents the same configuration as shown in (b), but with the three defects moved apart.

Figure 8

Optical image (dashed line) of the one-half-fermion density profile on top of a uniform background density with the average number of 0.5 atoms per lattice site produced by a specific imaging system, as discussed in the text. The density profile (solid line) that is being imaged is also shown. The size of the soliton is set by the choices $N_s=129$, $N_f=65$, and $\mu =-0.1a$, corresponding to the correlation length of $\xi \simeq 10$ lattice sites.

Figure 9

The intensity of light scattered from the optical lattice if the fermion numbers did not fluctuate, ${\left|⟨\widehat{N}⟩\right|}^2$, and the additional intensity due to fermion number fluctuations ${\left(\Delta N\right)}^2$, as a function of the size of the focus $w$ of the driving light given in the lattice units. We also display the added intensity ${\left(\Delta N\right)}_i^2$ that would result if fermion number fluctuations were uncorrelated between adjacent lattice sites. The soliton parameters are $N_s=129$, $N_f=65$, $\mu =-0.9a$.