$\epsilon$-pseudoclassical model for quantum resonances in a cold dilute atomic gas periodically driven by finite-duration standing-wave laser pulses

Atom interferometers are a useful tool for precision measurements of fundamental physical phenomena, ranging from the local gravitational-field strength to the atomic fine-structure constant. In such experiments, it is desirable to implement a high-momentum-transfer “beam splitter,” which may be achieved by inducing quantum resonance in a finite-temperature laser-driven atomic gas. We use Monte Carlo simulations to investigate these quantum resonances in the regime where the gas receives laser pulses of finite duration and derive an $\epsilon$-classical model for the dynamics of the gas atoms which is capable of reproducing quantum resonant behavior for both zero-temperature and finite-temperature noninteracting gases. We show that this model agrees well with the fully quantum treatment of the system over a time scale set by the choice of experimental parameters. We also show that this model is capable of correctly treating the time-reversal mechanism necessary for implementing an interferometer with this physical configuration and that it explains an unexpected universality in the dynamics.

Figure 1

Schematic of a possible experimental setup [37]. The time-of-flight (TOF) beam measures the atomic momentum distribution. If it is vertically oriented, the effect of the gravitational field can be transformed away [38] by use of the phase shifter element, for example, an electro-optic modulator [37].

Figure 2

Poincaré sections for $(\theta ,\mathcal{J}(\beta \left)\right)$ as evolved by Eq. (13) and Eq. (14), corresponding to the (a) $\beta =0$, (b) $\beta =0.05$, (c) $\beta =0.2$, and (d) $\beta =0.25$ subspaces, with $\ell =2$ and $\tilde{V}=0.251$. Each black circle represents one of 100 initial phase-space points, and each color represents the evolution of a single phase-space point over 1000 kicks. The smaller black points in (c) and (d) link up the rotational or elliptic orbits, respectively (which for $\beta =0.2$ and $\beta =0.25$, respectively, takes substantially longer than 1000 kicks).

Figure 3

Poincaré sections for $(\theta ,\mathcal{J}(\beta \left)\right)$ as evolved by Eq. (13) and Eq. (14), corresponding to the $\beta =0$ subspace for driving strengths (a) $\tilde{V}=0.251$, (b) $\tilde{V}=2.51$, (c) $\tilde{V}=5.01$, and (d) $\tilde{V}=7.51$, with $\ell =2$. Each black circle represents one of 100 initial phase-space points, and each color represents the evolution of a single phase-space point over 1000 kicks.

Figure 4

(a) Plot of $\langle {\widehat{p}}^2\rangle$ in units of ${\hslash }^2{K}^2$ vs number of kicks for a zero-temperature gas, with ${\varphi }_d=0.8\pi$ and $\ell =2$. The scaled pulse duration $\epsilon$ takes the values ${10}^{-2+2j/10}$, where $j=\left\{0,1,2,\cdots ,9\right\}$. Curves represent the results of the quantum dynamics [Eq. (5)], and points those of the $\epsilon$-pseudoclassical model [Eqs. (13) and (14)], with lower values of $\epsilon$ giving rise to higher peak values of $\langle {\widehat{p}}^2\rangle$. Hence, the uppermost curve (black) corresponds to $\epsilon =0.01$ ($j=0$) and the lowermost curve (red) to $\epsilon =0.631$ ($j=9$). (b) Rescaling of (a) by ${\epsilon }^2$ on the $\langle {\widehat{p}}^2\rangle$ axis and $\epsilon$ on the kick-number axis such that a universal curve is revealed, where all data overlap over a suitably short time scale.

Figure 5

Plots of the time evolution of the log of $\langle {\widehat{p}}^2\rangle$, in units of ${\hslash }^2{K}^2$ vs number of kicks, for different values of the quasimomentum $\beta =\{0,0.05,0.1,0.15,0.2,0.25\}$, for an otherwise zero-temperature gas [initial momentum eigenstate with $\mathcal{J}\left(\beta \right)=0]$. Smooth curves represent results of the quantum evolution [Eq. (5)], and points those of the effective classical model [Eqs. (13) and (14)]. In (a) $\epsilon =0.001$, and in (b) $\epsilon =0.2$. Other parameters are ${\varphi }_d=0.8\pi$ and $\ell =2$.

Figure 6

Comparison of the dynamics of the momentum distributions computed by the fully quantum model [Eq. (5)] vs the pseudoclassical model [Eqs. (13) and (14)] for zero-temperature ($w=0$) and finite-temperature ($w=2.5$) gases, with ${\varphi }_d=0.8\pi$ and $\ell =2$, for differing values of the scaled pulse duration $\epsilon$. (a–f) Momentum distributions for a zero-temperature gas ($w=0$) as computed by the quantum (a, c, e) and pseudoclassical (b, d, f) models. (h–m) Momentum distributions computed by the quantum (h, j, l) and effective classical (i, k, m) models, for $w=2.5$. In each row, the distribution dynamics are computed for a different value of $\epsilon$: for row 1 (a, b, h, i) $\epsilon =0.02$, for row 2 (c, d, j, k) $\epsilon =0.11$, and for row 3 (e, f, l, m) $\epsilon =0.2$. To accommodate the logarithmic color scale, we have chosen a cutoff value of $C={10}^{-11}$. The corresponding time evolution of $\langle {\widehat{p}}^2\rangle$ (in units of ${\hslash }^2{K}^2$) for (g) $w=0$ and (n) $w=2.5$; solid lines represent results of quantum calculations, and symbols results of the effective classical model (squares correspond to $\epsilon =0.2$, triangles to $\epsilon =0.11$, and circles to $\epsilon =0.02$). Monte Carlo calculations were carried out with $N_c={10}^5$ particles, or $N_q={10}^5$ state vectors, as appropriate.

Figure 7

Comparison of the dynamics of the momentum distributions computed by the fully quantum model [Eq. (5)] vs the pseudoclassical model [Eqs. (13) and (14)] for zero-temperature ($w=0$) and finite-temperature ($w=2.5$) gases, with ${\varphi }_d=0.8\pi$ and $\ell =2$, for differing values of the scaled pulse duration $\epsilon$. In each case a time-reversal event (phase-shifitng the standing wave by $\pi$) occurs at the 15th of 30 kicks (marked by the dashed lines). Momentum distributions for a zero-temperature gas ($w=0$) as computed by (a, c, e) the quantum model and (b, d, f) the pseudoclassical model. Momentum distributions computed by (h, j, l) the quantum model and (i, k, m) the effective classical model, for $w=2.5$. In each row, the distribution dynamics are computed for a different value of $\epsilon$: for row 1 (a, b, h, i) $\epsilon =0.02$, for row 2 (c, d, j, k) $\epsilon =0.11$, and for row 3(e, f, l, m) $\epsilon =0.2$. To accommodate the logarithmic color scale, we have chosen a cutoff value of $C={10}^{-11}$. The corresponding time evolution of $\langle {\widehat{p}}^2\rangle$ [in units of ${\hslash }^2{K}^2]$ for (g) $w=0$ and (n) $w=2.5$; solid lines represent results of quantum calculations, and symbols results of the effective classical model (squares correspond to $\epsilon =0.2$, triangles to $\epsilon =0.11$, and circles to $\epsilon =0.02$). Monte Carlo calculations were carried out with $N_c={10}^5$ particles, or $N_q={10}^5$ state vectors, as appropriate.

Figure 8

Plots of the time evolution of $\langle {\widehat{p}}^2\rangle /w^2$ (in units of ${\hslash }^2{K}^2$) vs the number of kicks, with $\epsilon =0.2, {\varphi }_d=0.8\pi$, and $\ell =2$. Each set of points corresponds to an individual value of $w={10}^{-1+2j/5}$, where $j=\left\{0,1,2,\cdots ,5\right\}$, as computed by the pseudoclassical model [Eqs. (13) and (14)].