Synthetic dimensions for cold atoms from shaking a harmonic trap

We introduce a simple scheme to implement synthetic dimensions in ultracold atomic gases, which only requires two basic and ubiquitous ingredients: the harmonic trap, which confines the atoms, combined with a periodic shaking. In our approach, standard harmonic oscillator eigenstates are reinterpreted as lattice sites along a synthetic dimension, while the coupling between these lattice sites is controlled by the applied time modulation. The phase of this modulation enters as a complex hopping phase, leading straightforwardly to an artificial magnetic field upon adding a second dimension. We show that this artificial gauge field has important consequences, such as the counterintuitive reduction of average energy under resonant driving, or the realization of quantum Hall physics. Our approach offers significant advantages over previous implementations of synthetic dimensions, providing an intriguing route towards higher-dimensional topological physics and strongly-correlated states.

Figure 1

(a) A particle in a periodically shaken harmonic trap [Eqs. (1, 2, 3)] can be understood as moving along a synthetic dimension indexed by the HO number $\lambda$ [Eq. (9)]. The modulation phase $\varphi$ enters as a complex hopping phase factor, and so can be used to create artificial magnetic fields when more dimensions are added as in (b). In this configuration, an atomic cloud initially prepared in the lowest-lying harmonic oscillator states ($\lambda \approx 0$) undergoes a unidirectional motion along the “real” dimension $y$, until it reaches the edge at $y=0$; after reaching this point, the motion along $y$ stops, and the average energy increases as the atoms are driven to higher harmonic oscillator states. This motion is analogous to the unidirectional propagation of chiral edge states in 2D quantum Hall systems.

Figure 2

Comparison between the tunneling matrix elements $|J_{\lambda }|$ of the Floquet Hamiltonian in Eq. (11) (red dots), and the analytical expression (7) associated with the effective Hamiltonian in Eq. (9) (blue line). The breakdown of the rotating-wave approximation (8) is visible when $|J_{\lambda }|\gtrsim \omega /4$. The numerical data were obtained by splitting the time-evolution operator $\widehat{U}(T_{\text{D}};0)$ into 200 time steps, each of duration ${\delta }_t=T_{\text{D}}/200$, and within which the Hamiltonian was assumed to be constant; after evaluating $\widehat{U}(T_{\text{D}};0)$, the Floquet Hamiltonian matrix is obtained through the expression ${\widehat{H}}_F=i/T_{\text{D}}ln\left[\widehat{U}(T_{\text{D}};0)\right]$, which can be numerically evaluated. Here, we set the detuning $\mathrm{\Delta }=0$.

Figure 3

Numerical COM evolution ${\lambda }_{\text{c.m.}}\left(t\right)$ of a Gaussian wave packet for the 1D lattice model [Eq. (9)] (blue line), for the full Hamiltonian $\widehat{H}\left(t\right)={\widehat{H}}_0+\widehat{V}\left(t\right)$ (red circles) and for the analytical classical COM evolution [Eq. (18)] (black line), under a “force” aligned along the synthetic dimension, as generated by the modulation detuning $\mathrm{\Delta }$. (a) When $\mathrm{\Delta }=0$, there is a runaway increase in ${\lambda }_{\text{c.m.}}\left(t\right)$ and hence of the average energy $E_{\text{av}}\left(t\right)=\omega {\lambda }_{\text{c.m.}}\left(t\right)$. (b) For small detuning, here $\mathrm{\Delta }=0.1\omega$, there are large Bloch oscillations. In both panels, $\kappa =0.1\omega /l_{\text{H}}, \varphi =0$, and the wave packet is initialized with ${\lambda }_0=20$ and $\sigma =5$. Time is expressed in HO periods $T=2\pi /\omega$.

Figure 4

Classical COM evolution of a Gaussian wave packet up to $t=20T$, as described by Eq. (16), for two values of the modulation detuning $\mathrm{\Delta }$. (a) Unstable trajectory in the case $\mathrm{\Delta }=0$; this is to be compared with the diverging behavior of ${\lambda }_{\text{c.m.}}\left(t\right)$ in Fig. 3. (b) Stable and periodic trajectory when $\mathrm{\Delta }=0.1\omega$; this corresponds to large Bloch oscillations in $\lambda$ space [Fig. 3]. Parameters are the same as in Fig. 3.

Figure 5

(a) Two lattice sites along a second dimension, as given by (left) two internal atomic states $|\uparrow ,\downarrow \rangle$, or by (right) a double well of spacing $a$ along $y$. Alternatively, the two lattice sites can correspond to two suitably coupled microtraps along $x$ (see Sec. 4). (b) Combined with the synthetic dimension $\lambda$, such a system can be viewed as a two-leg ladder pierced by a uniform magnetic flux $\mathrm{\Phi }={\varphi }_{1/2}-{\varphi }_{-1/2}$, imposed by spin-dependent modulation phases ${\varphi }_{-1/2,1/2}$. (c) With more sites along the second dimension, e.g., from an optical lattice along $y$, the system acts like a 2D (anisotropic) HH model.

Figure 6

(a) Spectrum for an isotropic two-leg ladder with constant hoppings $J_{\lambda }=2J_y$ and $\mathrm{\Phi }=\theta =\pi /2$, where the arrows $(\uparrow ,\downarrow )$ and colors are both used to specify the mean spin. (b), (c) Numerical time evolution using the full Hamiltonian $\widehat{H}\left(t\right)={\widehat{H}}_0+{\widehat{H}}_y+\widehat{V}\left(t\right)$. The wave packet is initialized at (b) $m=1/2$ and (c) $m=-1/2$, with ${\lambda }_0=20$ and $\sigma =5$. Spin-resolved and total densities are indicated in the three successive panels. Other parameters are $\mathrm{\Phi }=\theta =\pi /2, \kappa =0.01\omega /l_{\text{H}}, J_y=J_{{\lambda }_0}/2\approx 8\times {10}^{-3}\omega , \mathrm{\Delta }=0$, so as to resemble the isotropic model in (a). Due to the chiral motion, the average energy only increases beyond its initial value after a time $t\approx 550T$.

Figure 7

Center-of-mass trajectory ${\lambda }_{\text{c.m.}}\left(t\right)$ obtained from a direct numerical evaluation of the time-evolution operator $\widehat{U}(T_{\text{D}};0)$ associated with the full time-dependent Hamiltonian (26) (red dots), and compared with the trajectory generated using the effective Hamiltonian (27). Here the detuning is $\mathrm{\Delta }=0$, hence $T=T_{\text{D}}=2\pi /\omega$, and all other parameters are given in the caption of Fig. 6. Note that in both cases, the time evolution is probed stroboscopically, at times given by $t=N{T}_{\text{D}}$, with $N$ integer (i.e., dynamics are generated by ${\left[\widehat{U}(T_{\text{D}};0)\right]}^N$, with $N=1,\cdots ,700$).

Figure 8

(a) Spectrum of the 2D anisotropic HH model for periodic boundary conditions along $y$, and $\lambda \in [0,80]$ with open boundaries; the color scale indicates the weight along $\lambda$, highlighting states with $\langle \lambda \rangle <30$. The group velocity of the lowest chiral edge mode, at quasimomentum $q_y\approx 0.8/a$, is indicated. (c) Full time evolution of a wave packet [Eq. (36)], prepared around $m\approx 60$ and ${\lambda }_0\approx 0$, with mean momentum ${\overline{q}}_y=0.8/a$, and widths given by ${\sigma }_{\lambda }=1$ and ${\sigma }_y=7a$; see Eq. (36). Parameters are $\mathrm{\Phi }=\pi /2, \kappa =0.01\omega /l_{\text{H}}, J_y=0.02\omega$, and $\mathrm{\Delta }=0$. The colorbar indicates the particle density $\rho (m,\lambda )$, as defined in the $m-\lambda$ plane.

Figure 9

Generating an extra dimension, using a time-modulated optical superlattice. (a) Two neighboring harmonic wells, denoted 1 and 2, respectively. The coupling matrix elements between harmonic-oscillator eigenstates associated with the two wells are denoted $J_{\lambda ,{\lambda }^{\prime }}^x$. (b) Introducing an energy offset ${\mathrm{\Delta }}_{\text{off}}$ between the wells and a time modulation of the wells' depth [65, 66] allows one to control the hopping between the wells, resulting in the effective Hamiltonian in Eq. (40). (c) The effective Hamiltonian in Eq. (40) describes “diagonal” hopping along $x$ and along $\lambda$.

Figure 10

Generating an extra dimension, using two state-dependent optical lattices.

Figure 11

(a) Schematic of two particles at ${\lambda }_1={\lambda }_2=20$ scattering into different states ${\lambda }_3=20+n$ and ${\lambda }_4=20-n$. (b) The corresponding relative interaction strength $U({\lambda }_1,{\lambda }_2;{\lambda }_3,{\lambda }_4)$ as a function of $n$.

Figure 12

(a) Schematic of two particles at $({\lambda }_1,{\lambda }_2)=(20+n,20-n)$ scattering into the same states $({\lambda }_3,{\lambda }_4)=(20+n,20-n)$. (b) The corresponding relative interaction strength $U({\lambda }_1,{\lambda }_2;{\lambda }_3,{\lambda }_4)$ as a function of $n$.

Figure 13

(a) Schematic of two-particle scattering when ${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_4=n$. (b) The corresponding relative interaction strength $U({\lambda }_1,{\lambda }_2;{\lambda }_3,{\lambda }_4)$ as a function of $n$.

Figure 14

The current in each leg of the ladder, calculated from the eigenstates of the effective Hamiltonian in Eq. (27), using Eq. (A2). Results are shown for (a) an isotropic two-leg ladder (i.e., setting $J_{\lambda }=J=1$), and for (b) the anisotropic two-leg ladder [Eq. (27)]; blue circles and red crosses correspond to ${\langle {\widehat{j}}_{\uparrow }\rangle }_n$ and ${\langle {\widehat{j}}_{\downarrow }\rangle }_n$, respectively [see Eqs. (A1) and (A2)]. In each case, we have diagonalized the Hamiltonian for a region of 201 lattice sites along each leg, corresponding in the anisotropic model to $\lambda \in \{0,\cdots ,200\}$. We also take $\mathrm{\Phi }=\pi /2$ and $\theta =0$ and all quantities are expressed in terms of the energy scale $J_y$, which is set to $J_y=J/2$ in the isotropic limit and $J_y=J_{\lambda =200}/2$ in the anisotropic case. Note how the anisotropy distorts the structure in (a), but still presents energy-dependent chiral behavior in reasonably large energy ranges.