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Conduction of Ultracold Fermions Through a Mesoscopic Channel

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Science  31 Aug 2012:
Vol. 337, Issue 6098, pp. 1069-1071
DOI: 10.1126/science.1223175

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Pretend Wires

Cold atomic gases have been successfully used to simulate solid-state phenomena such as quantum criticality. However, simulating mesoscopic electronic transport like that realized in quantum wires is challenging. Brantut et al. (p. 1069, published online 2 August) connected two reservoirs of fermionic 6Li atoms (simulating electrons) with a narrow channel (simulating a wire), created a nonequilibrium situation by applying a magnetic field gradient, and observed the flow through the channel. When the mean-free path of the atoms exceeded the length of the channel, the atomic density in the channel was constant in the central region and only changed at the ends, indicating the presence of contact resistance. The opposite diffusive regime created by imposing a disordered laser potential produced a uniformly varying density inside the channel.

Abstract

In a mesoscopic conductor, electric resistance is detected even if the device is defect-free. We engineered and studied a cold-atom analog of a mesoscopic conductor. It consists of a narrow channel connecting two macroscopic reservoirs of fermions that can be switched from ballistic to diffusive. We induced a current through the channel and found ohmic conduction, even when the channel is ballistic. We measured in situ the density variations resulting from the presence of a current and observed that density remains uniform and constant inside the ballistic channel. In contrast, for the diffusive case with disorder, we observed a density gradient extending through the channel. Our approach opens the way toward quantum simulation of mesoscopic devices with quantum gases.

The quantum simulation of models from solid-state physics using cold atoms has seen tremendous progress over the past decade (1, 2). Still, there are only limited analogies to the concept of conduction, which is at the core of mesoscopic solid-state physics. To close this gap, it would be highly desirable to connect a probing region in a cold-atom experiment to external incoherent reservoirs. This would lead to directed transport, the control of which is the basis of electronics. In such an intrinsically open configuration, boundary conditions play a crucial role, as in the Landauer theory of transport (3). The transport properties in cold atom systems have been investigated by observing the response of the system to variations of the external potential (413) or by monitoring the coherent evolution of bimodal Bose-Einstein condensates (1416) as a response to a bias. Extending the concept of quantum simulation to conduction requires the engineering of macroscopic reservoirs, an atom battery or capacitor connected to the conductor (1719).

We report on the observation of atomic conduction between two cold-atom reservoirs through a mesoscopic, multimode channel. Our measurement is made possible by the separation of scales in our trap geometry (Fig. 1). The experimental configuration consists of two identical, macroscopic cold-atom reservoirs, which contain the majority of the atoms and feature fast equilibration dynamics. They are connected by a channel that contains a negligible fraction of the atoms and supports a few quantum states in the z direction, while it has the same extension as the reservoirs in the x direction, making it quasi–two-dimensional (quasi-2D).

Fig. 1

Experimental configuration. A macroscopic 6Li cloud was divided into two reservoirs separated by a narrow channel. The channel was imprinted by using the two lobes in the intensity profiles of a nearly TEM01–mode laser beam at the wavelength of 532 nm, created with a holographic plate. The distance between the two lobes is 18 μm, and the waist of the beam in the y direction is 30 μm. A microscope objective [numerical aperture (NA) = 0.55] was used to observe or manipulate the atoms in the channel. When NR is larger than NL, an atomic current I flows through the channel (arrow).

We first prepared quantum degenerate gases containing Ntot = 4 × 104 6Li atoms in each of the two lowest hyperfine states at a temperature of 0.36 ± 0.18 [0.36(18)] TF, where TF ≈ 700 nK is the Fermi temperature in a combined optical and magnetic trap (20). A laser beam propagating along the x direction was focused on the center of the atomic cloud. The beam had a nodal line in the middle of its intensity profile and produced a repulsive potential for the atoms, which is tightly confining in the z direction (21, 22). Oscillation frequencies of up to 3.9 kHz along the z direction were achieved (Fig. 1).

Figure 2A presents a typical absorption picture of a cloud in the presence of the channel. We observed two clouds separated by a low density region, revealing the presence of the channel and confirming that it contains a negligible fraction of the total atom number (smaller than 0.01).

Fig. 2

Atomic reservoirs connected by a mesoscopic channel. (A) Absorption image of the atoms before the imbalance is applied. The image was taken after a 1-ms expansion in the x-z plane. The dark region at the center reveals the presence of the mesoscopic channel. The dashed line tracks the position of the channel on both panels. (B) Density difference between the cloud in the unbalanced configuration, as the current sets in, and the equilibrated cloud. Excess of atoms is displayed in orange, and lack of atoms is displayed in green. The imbalance of the reservoirs, ∆N/Ntot, was set to 0.2.

The conduction measurement proceeded as follows. We created an asymmetry in the potential by applying a constant magnetic field gradient of 2.5 mT m−1 along the y axis. This was done during the evaporation process and eventually resulted in an imbalance ∆N/Ntot ≈ 0.2, where ∆N is the number difference between right and left reservoirs. After evaporation, the confining potential of the trap was increased, and a uniform magnetic field was set to 47.5 mT. At this value, the scattering length of atoms in the two internal states is –100 a0, with a0 being the Bohr radius. This ensures that the collision rate is sufficient to maintain thermal equilibrium in each reservoir on a time scale of ~30 ms. It also ensures that the mean free path (~1.3 mm) is much larger than the length of the channel, making it ballistic. The symmetry of the trapping potentials was then restored by switching off the magnetic field gradient in 50 ms, a time longer than the internal thermalization time of each reservoir but short compared with the time scale of equilibration of the populations of the two reservoirs. Figure 2B shows the difference between an absorption picture taken with and without imposing an imbalance. The right reservoir is seen to contain an excess of particles compared with the balanced reservoirs situation, and the left reservoir shows a deficit of particles.

The equilibrium of the whole system is characterized by a balanced population of both reservoirs; thus, after restoring the symmetry of the trap, an atomic current sets in through the channel. Figure 3A presents the time evolution of ∆N/Ntot, with the oscillation frequency along z in the channel set to 3.9 kHz. We observed an exponential decay (Fig. 3A, solid line), with a time constant of 170 ± 14 [170(14)] ms. This exponential shape suggests a direct analogy with the discharge of a capacitor through a resistance. Indeed, the evolution of the system can be described as ddtΔN=GCΔN (1)where G is the conductance of the channel, C = ∂N/∂μ is the compressibility of the reservoirs, and μ is their chemical potential. The compressibility is analogous to the capacity of a capacitor. We neglected possible thermoelectric effects, because we did not observe a noticeable temperature evolution in the reservoirs.

Fig. 3

Observation of ohmic conduction. (A) Measured number difference between the two reservoirs as a function of time. The solid line is an exponential fit to the data. (B) Current as a function of number difference between the two reservoirs, measured from the exponential fit of (A), for two different confinements in the channel. A small offset obtained from the fits in (A), which is due to a slight misalignment of the channel with respect to the center of the trap, has been substracted. Circles, maximum center frequency along z set to 3.9 kHz; triangles, 3.2 kHz. The lines are linear fits to the data. Error bars indicate statistical errors.

Because the decay is the slowest process, we have a quasi–steady-state situation at each point in time. Thus, the derivative of the curve around any point is a measurement of the current at a certain number difference, where the atoms in each reservoir have a known, thermal distribution. Therefore, the magnitude of the current measures the dc characteristic of the channel. Figure 3B shows the observed current as function of the number difference for the same data set (circles) and for a channel with reduced confinement of 3.2 kHz at the center (triangles). A linear relation is manifest for both cases, which confirms dissipative, ohmic conduction and allows us to extract the slopes G/C = 2.9(4) s−1 and 3.7(2) s−1, respectively.

The observation of resistance in the ballistic conduction shows that the boundary conditions are essential in the investigation of transport, as in the Landauer approach. Indeed, the free expansion of a noninteracting cloud is also ballistic, but the absence of connection to reservoirs leads to the absence of any resistance to the flow, other than inertia. Furthermore this ballistic expansion generally does not depend on the conduction properties of the initial cloud, because ballistic expansion has even been observed for a band insulator (12).

The Landauer-Büttiker formula states that, at zero temperature, the conductance of a ballistic conductor is equal to 1/h per quantum state contributing to the conduction, where h is Planck’s constant (23). Because of the quasi-2D character of the channel, the current is carried by many transverse modes, which are not individually resolved because of finite temperature. Instead, as the channel confinement is varied, the resistance is expected to vary linearly with the oscillation frequency along the confined direction z, because of the variations in the number of available modes. In both measurements, the reservoirs have the same compressibilities; thus, the ratio 0.76(11) of the two slopes is equal to the ratio of conductances alone and agrees qualitatively with the inverse ratio 0.82 of trap frequencies along z. We found that the linear relation between resistance and trap frequency persists for various confinements within the accessible range. The contact resistance, which naturally appears in the Landauer picture, explains the observation of ohmic conduction even in a defect-free channel. Although every atom that enters the channel on one side exits on the other with the same momentum with probability one, only a tiny fraction of the atoms from each reservoir can pass through the channel at any given time because of the low density of states in the channel (24, 25).

To gain further insight into this mechanism, we used high-resolution microscopy to observe the density distribution of atoms in the channel. We did so by using in situ absorption imaging along the z direction, with and without current flowing through the channel. A typical picture of the density distribution in the channel in the absence of current is presented in Fig. 4A. At the sides of the picture, we observe the contacts with the two reservoirs that extend beyond the field of view. Closer to the center, the lower column density reveals the presence of the channel, which is smoothly connected to the reservoirs. Figure 4B shows the difference between two such pictures, taken with and without current flowing through the channel. We see the small density difference between the two reservoirs, which reflects the macroscopic number difference shown in Fig. 2B.

Fig. 4

Investigation of conduction using high-resolution microscopy. (A) Absorption picture of the density in the channel for a cloud at equilibrium (no current). (B) Difference between two pictures taken at equilibrium and with a current of 104 s−1. The color is orange for positive difference and green for negative. (C) Line-density difference obtained by accumulating (B) along the x axis, for a ballistic channel (red solid circles) and for a diffusive channel having the same conductance (blue open circles), in the presence of the same current. The solid lines are smoothed data to guide the eye. (D) Focus on the central part of the line-density difference. (E) Resistivity in the channel for the diffusive case, as a function of position, computed from Eq. 2. Only the center of the channel is shown, where the extraction procedure for the line compressibility is valid (28). (F) Mobility in the channel as a function of position for diffusive channel. The error bars and shaded region reflect the uncorrelated combination of estimated systematic and statistical uncertainties.

The red points in Fig. 4C show the line-density difference ~nl along the channel, obtained by accumulating the image in Fig. 4B along the x direction. At the center of the channel, the difference is close to zero over a length of 30 μm, whereas the density difference changes quickly at the sides of the channel. This qualitative difference between the center of the channel and the sides indicates that the resistance observed in Fig. 3 originates from the reflection of atoms by the contacts with the reservoirs (26).

As opposed to the ballistic channel, we have engineered a channel where the conduction is diffusive, which is the case encountered in typical solid materials. To do so, we projected a blue-detuned laser speckle pattern onto the channel, realizing a quasi-2D disorder (27). This pattern has a gaussian envelope with a root mean square diameter of 32 μm, an average amplitude of 0.6 μK at the center, and a correlation radius of 0.37 μm (28). We then reduced the confinement of the channel down to 1.6 kHz along z, so that the atomic conductance of the disordered channel was the same as that of the ballistic one studied before. We thus have a second system displaying the same macroscopic transport properties but with a different conduction mechanism. The measured line-density difference in the disordered channel is shown in blue in Fig. 4C (blue dots). In sharp contrast to the ballistic case, the density difference exhibits a continuous decrease from right to left.

For the diffusive transport case, we now relate the variations of density difference to local transport quantities. Following the approach of mesoscopic physics (25), we introduced an effective local chemical potential, μ(y), by requiring that it yields the observed density when used in the Fermi-Dirac distribution. Because the mean free path for atomic collisions is very large, the energy distribution of atoms may not be thermal outside the reservoirs. Therefore, the definition of the effective chemical potential μ(y) does not correspond to the local density approximation and does not suppose local equilibrium.

From the current I measured across the channel, we deduce from Fig. 4C the local resistivity (28)ρ=1κ1In˜1y (2)where we have introduced the line compressibility in equilibrium, κl = ∂nl/∂μ, with nl being the line density along the y axis. The line compressibility was obtained directly from the column density at equilibrium and the shape of the trap (28, 29). Figure 4E presents the resistivity obtained by applying Eq. 2 to the in situ picture. It remains finite across the channel and presents two weak maxima, which we attribute to the fastest variations of the confining potential creating the channel. The local resistance in the channel has its origin in the scattering with the random potential, which leads to randomization of the momentum distribution of the atoms (25).

Many quantities of interest can be extracted from the microscopic density distribution. For instance, the drift velocity, vd = I/nl, in the channel is found to be 200 μm s−1, or 4 × 10−3 vF, where vF is the Fermi velocity in the reservoirs, which confirms that our system realizes the Laudauer paradigm of conduction. For the diffusive channel, we also introduced an atomic mobility for the atoms, vdκ(∂nl/∂y)−1, which relates the drift velocity to the effective chemical potential gradient and thus characterizes the intrinsic conduction properties of the channel regardless of the density. Figure 4F presents the atomic mobility as obtained from the in situ pictures for the diffusive channel, which remains finite through the channel, with a weak maximum at the center.

Our configuration is closely analogous to that of a field-effect transistor. The strength of the confinement in the channel has been used to vary the conductance by changing the density. Further tuning of the resistance could be obtained by adding a repulsive gate laser. In addition, the effects of disorder in such a device can be studied systematically by varying the laser-induced random potential. Metal-insulator transitions, such as two-dimensional Anderson localization (30), can be studied in a way that is directly analogous to real solid-state devices (31). The ability to further control the disorder could be used to study universal conductance fluctuations (3). Apart from disorder, various potentials can be designed and projected onto the channel using the microscope setup (20). This will allow us to measure the conduction properties of various model systems. For example, quantized conduction can be investigated if a single mode can be resolved in the channel (3234). Furthermore, conductance is very sensitive to interactions between atoms and would be an ideal observable to investigate strongly correlated fermions. The combination of mesoscopic atomic devices with controlled interactions opens fascinating perspectives and could shine new light on open questions in the field of mesoscopic physics (35).

Supplementary Materials

www.sciencemag.org/cgi/content/full/science.1223175/DC1

Materials and Methods

Fig. S1

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We acknowledge fruitful discussions with J. Blatter, A. Georges, H. Moritz, A. Rosch, and W. Zwerger and the help of T. Müller during the early stage of the experiment. We are grateful to the group of F. Merkt for the loan of a 532-nm laser. We acknowledge financing from National Centers for Competence in Research Materials with Novel Electronic Properties (NCCR MaNEP) and Quantum Science and Technology (QSIT), European Research Council (ERC) project Synthetic Quantum Many-Body Systems (SQMS), Framework Program 7 (FP7) project Nanodesigning of Atomic and Molecular Quantum Matter, and ETH Zürich. J.P.B. acknowledges support from the European Union (EU) through a Marie Curie Fellowship.
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