At last, we are in the position to discuss the final stage of the experiment, namely the in-situ imaging of the atomic distribution. We deploy conventional absorption imaging technique. We first outline the modified Beer-Lambert law used to deduce the optical density of the sample.
3With an upgraded arbitrary waveform generator, we can now generate flat-top pulse in larger time window of Tpulse= 14 ms for sharper frequency response.
The modified Beer-Lambert law To image the atoms, we make use of the D2transition of with
σ−-polarised light (see Fig.3.2
b). The Beer-Lambert law describes the degradation of light intensity when propagating through a dilute medium of density n(x, y, z) is given by [89],
dI(x, y, z)
dz = −σ0n(x, y, z)I(x, y, z). (3.19)
Here, σ0is the scattering cross section of two-level atoms in the vicinity of the D2transition and z is the along the light propagating direction. Integrating Eq. (3.19) along the z direction results in an optical density based on logarithm of fraction of absorbed light intensity, i.e.
σ0n(x, y) = − ln I f(x, y) Ii(x, y) . (3.20)
However, due to saturation effects in Eq. (3.21), the photon scattering cannot take place efficiently in the case of high imaging intensity, resulting in the intensity-dependent scattering cross section
σ(I) = σ0
1 + (2∆ Γ)
2+ s. (3.21)
Similar to the notations used in Sec. 3.3.1, ∆ = ω − ω0is the detuning of imaging light with respect to the transition frequency ω0. Γ is the natural linewidth of the D2line and the saturation parameter s = I/Isat is the ratio of the imaging intensity to the saturation intensity. The relevant properties of the D2transition are provided in Table. 3.1. Eq. (3.19), therefore, needs to be modified with intensity-dependent scattering cross section which takes into account the effect of saturation. With this in mind, the modified Beer-Lambert law reads [90]
σ0n(x, y) = −α ln I f(x, y) Ii(x, y) + Ii(x, y) − If(x, y) Isat0 . (3.22)
The calibration procedure for obtaining effective camera counts corresponding to Isat0 can be found in [73]. To this end, we deploy an imaging pulse of 10 µs, with intensity I ≈ 1.5Isat. The imaging setup along the vertical direction is shown in Fig.3.13.
Frame transfer scheme To record multiple images in a single experimental run within a short
period of time, we deploy the fast kinetics mode of the Andor iXon888 camera. The camera contains a CCD sensor with resolution of 1024 pixel × 2048 pixel. Only an area of 1024 pixel × 512 pixel is illuminated, where the rest of the chip serves as storage area as shown in Fig.3.13b. Upon triggering, the CCD is exposed to imaging light for a time Texp. After this, the image acquisition pauses, and the recorded charges are shifted vertically by 512 pixels such that a new image can be recorded. This process repeats twice, giving access to two additional images. To this end, three consecutive exposures are taken.
Before recording the images, we have the to-be-recorded atomic cloud typically residing in the two lowest hyperfine states |F = 9/2, mF = −9/2i and |F = 9/2, mF = −7/2i. A microwave pulse is first shone onto the cloud to transfer the atoms in the |F = 9/2, mF = −9/2i state to the F = 7/2 manifold, which is insensitive to the imaging light used later. Next, the atoms in the |F = 9/2, mF = −7/2i state is transferred to the |F = 9/2, mF = −9/2i via a LZ sweep. Subsequently, the first exposure records the atomic ensemble in the |F = 9/2, mF = −9/2i with an imaging pulse duration of τimg= 10 µs.
z y x f = 200 mm To CCD camera f = 350 mm f = 8 mm 1024 pixels 2048 pixels pixels
Exposure 1 Exposure 2 Exposure 3
masked area
CCD chip
chip cleaning transfer transfer readout
MW amplitude RF amplitude CCD exposure imaging light
a
b
c
512Figure 3.13: Vertical imaging setup and fast kinetics scheme. a Main imaging setup along the z-direction.
bOutline of the CCD chip of the Andor iXon888 camera. c Details on the fast kinetics scheme. After each
exposure, the charges accumulated are shifted vertically down in 6.5 µs into the masked region for storage. b and c are adapted from [73].
After that, atoms in the F = 7/2 manifold are transferred back to a |F = 9/2, mF = −9/2i, followed by the second exposure. Finally, a third image is taken with no atoms remained in the system. This serves a “bright” image for computing the optical density using Eq. (3.22).4
4In addition the bright image, we typically record a series of images (up to ∼ 1000 reproductions) at which no light is shone onto the camera. This so-called dark image is then used in the post-analysis to eliminate the effect of stray charges accumulated during the image acquisition.
Simulating the Metal/Mott-insulator crossover
using attractive interaction
So our problem is to explain where symmetry comes from. Why is nature so nearly symmetrical?
Richard P. Feynman
In this chapter, we investigate the symmetry of the two-dimensional Hubbard model under a particle- hole transformation using the quantum simulator apparatus in our experiment. We first begin with a pedagogical discussion on various symmetries in the Hubbard model and its consequences. In particular, we focus on the particle-hole symmetry and the resulting correspondence in phase diagrams of the repulsive and attractive interaction. In order to discuss the physical behaviour of a strongly correlated Hubbard system, we introduce the concept of the equation of state in the Hubbard model and its experimental measurement. In particular, we experimentally observe a “spin-incompressible” phase with attractive interaction, equivalent to a Mott insulator under repulsive interaction. The results presented here validate a novel pathway towards simulating the Hubbard Hamiltonian. The experimental results discussed in this chapter was published in Phys. Rev. Lett. 124 (1), 010403 (2020).
4.1 Symmetries in the Hubbard model
In simple terms, a physical system is said to possess a symmetry when it remains unchanged under a specific transformation. The celebrated Noether’s theorem states that every continuous, differentiable symmetry of a physical system corresponds to a conservation law [91]. One example is a uniform, rotating sphere which possesses global rotational symmetry. This implies that there is no bias in any direction. As a result, the rotation possesses the same angular momentum, regardless of any global rotated observing frame. Such Indistinguishability in direction leads naturally to the inability to describe the system differently along differently-rotated coordinates. Table4.1displays several common symmetries and their corresponding conversation laws and indistinguishabilities.
For a quantum system, the symmetry operations can be identified via a transformation of the Hamiltonian. For a Hamiltonian to fulfil a given symmetry, characterised by the operator ˆR, it needs to be invariant under the operation,
ˆH → ˆR† ˆH ˆR. (4.1)
It is not difficult to see that, when a physical system is invariant under certain symmetry operation, it can be significantly simplified. Most notably, the measurements (expectation values) would also be invariant under the transformation.1 In this section, we discuss the role of symmetry in the Hubbard model and state explicitly the symmetry operation and their corresponding consequences in our quantum simulator.
Symmetry Operation Conservation law Indistinguishability
Translation r → r+ δr momentum absolute space
Time-translation t → t+ δt energy absolute time
Rotation r → r0 angular momentum absolute direction
Space inversion r → −r parity absolute left/right
Time-reversal t → −t Kramers degeneracy absolute sign of time Charge inversion e → −e particle/anti-particles charge conjugation Particle exchange ψ(i, · · · , j) →
ψ(j, · · · ,i) Bose/Fermi Statistics identical particles
Table 4.1: List of symmetries in physical systems. Common continuous transformations including global translation, rotation and time-translation lead to the conservation of momentum, angular momentum and energy. Discrete symmetries such as parity, time-reversal and charge inversion form the combined CPT-symmetry which is observed in all physical laws so far. Under exchange of identical particles, the Bose-Einstein or Fermi-Dirac statistics arise depending on whether a symmetry or anti-symmetry is fulfilled.