Homodyne detection of matter-wave fields

A scheme is discussed that allows for performing homodyne detection of the matter-wave field of ultracold bosonic atoms. It is based on a pump-probe laser setup, which both illuminates a Bose-Einstein condensate, acting as reference system, and a second ultracold gas, composed by the same atoms but in a quantum phase to determine. Photon scattering outcouples atoms from both systems, which then propagate freely. Under appropriate conditions, when the same photon can either be scattered by the Bose-Einstein condensate or by the other quantum gas, both flux of outcoupled atoms and scattered photons exhibit oscillations, the amplitude of which is proportional to the condensate fraction of the quantum gas. The setup can be extended to measure the first-order correlation function of a quantum gas. The dynamics discussed here makes use of the entanglement between atoms and photons, which is established by the scattering process in order to access detailed information on the quantum state of matter.

Figure 1

Setup for performing homodyne detection of matter-wave fields. A gas of identical bosonic atoms is confined in two distinguishable regions of space. In the left region, it forms a Bose-Einstein condensate that acts as reference system for measuring the condensate fraction of the right system. A pump-probe laser setup outcouples the atoms from both wells. The photon flux of the probe beam is constituted by the photon scattered in the outcoupling process and exhibits oscillations, the amplitude of which depends on the condensate fraction of the right system, and which vanishes when this is zero. The first-order correlation function of a quantum gas can be determined in an extension of this setup and is discussed in Sec. 5. Details on the parameters are given in the text.

Figure 2

Photon flux $F\left(t\right)$ [in units of the background contribution $F_B\left(t\right)$] as a function of time (in units of $\delta \mu /2\pi$) and of the angle $\alpha$ at which the atoms are emitted. The photon flux is computed by numerically integrating Eq. (35). The condensates are composed by $N={10}^6$ sodium atoms in a spherical harmomic trap of frequency $\omega =325\mathrm{Hz}$. The scattering length is $a_s=55a_0$ with $a_0$ Bohr radius. The other parameters are $v_{\mathbf{q}}=6\frac{\mathrm{cm}}{\mathrm{s}}$, $\delta {\mu }_0=2\pi {10}^3\mathrm{Hz}$, $d=5r_x$, $\Omega ={\omega }_{\mathbf{q}}$, $\alpha =0$, and ${\varphi }_{LR}=0$.

Figure 3

(a) Interference term of the photon flux $F_I\left(t\right)$ [Eq. (39)] [in units of the background current obtained when both condensates are at zero temperature $F_B(T=0)$], as a function of time (in units of $2\pi /\delta \mu$) and temperature $T$ (in units of critical temperature $T_c^{\mathrm{i}}$). (b) Amplitude $\mathcal{C}\left(T\right)$ [Eq. (41)] [in units of $\mathcal{C}\left(0\right)$] as a function of the temperature $T$ (in units of $T_c^{\mathrm{i}}$). The red dashed line corresponds to the squared root of the condensate fraction in the right BEC, $\sqrt{n_C\left(T\right)}$. The other parameters are as in Fig. 2.

Figure 4

Setup for measuring the first-order correlation function of an ultracold atomic gas. The pump-probe lasers are tailored in such a way that atoms are only outcoupled from the two shaded regions around ${\mathbf{r}}_{1,2}$. Measurement of the photon flux of one beam as a function of $\mathbf{q}$ allows one to determine the first-order correlation function $G^{\left(1\right)}({\mathbf{r}}_{\mathbf{1}};{\mathbf{r}}_2)$ of the atomic gas.

Figure 5

Comparison of the integral $G\left(z\right)$ as given in Eq. (B4) (black solid line) with the Gaussian fit Eq. (B5) (red dashed line).

Figure 6

Total photon flux $F\left(t\right)$ in units of background contribution $F_B\left(t\right)$ as a function of time in units of $\frac{\delta \mu }{2\pi }$. The solid black line is computed by numerically integrating Eq. (35) and compared to the approximate result Eq. (36) (dashed red line). Parameters are as in Fig. 2 with $\alpha =0$.