Controlling and observing nonseparability of phonons created in time-dependent 1D atomic Bose condensates

We study the spectrum and entanglement of phonons produced by temporal changes in homogeneous one-dimensional atomic condensates. To characterize the experimentally accessible changes, we first consider the dynamics of the condensate when varying the radial trapping frequency, separately studying two regimes: an adiabatic one and an oscillatory one. Working in momentum space, we then show that in situ measurements of the density-density correlation function can be used to assess the nonseparability of the phonon state after such changes. We also study time-of-flight (TOF) measurements, paying particular attention to the role played by the adiabaticity of opening the trap on the nonseparability of the final state of atoms. In both cases, we emphasize that commuting measurements can suffice to assess nonseparability. Some recent observations are analyzed, and we make proposals for future experiments.

Figure 1

The chemical potential as a function of the adimensionalized linear densities, $n_1{a}_s$ and $n_1\xi$, in the Gaussian approximation; see Eq. (4). For clarity we have plotted ${\mu }^2/{\omega }_{\perp }^2-1$, which in this approximation is linear in $n_1{a}_s$. We have marked the region $n_1{a}_s\gtrsim 1$ (where the gas exhibits three-dimensional behavior) and the region $n_1\xi \lesssim 1$ (where the gas exhibits Tonks-Girardeau behavior [36]). To relate $n_1\xi$ to $n_1{a}_s$, see Eq. (7), we have used the value of ${(a_s/a_{\perp })}^2=3\times {10}^{-5}$ corresponding to the experiment of [15]. The two black dots (roughly) correspond to the subsonic ($n_1{a}_s\approx 0.4$) and supersonic ($n_1{a}_s\approx 0.05$) sides of the analogue black hole realised in [15], while the circle gives a rough indication of the position of the modulated DCE experiment of [11] ($n_1{a}_s\approx 0.2$).

Figure 2

The response of the condensate to a temporal change in the radial trap frequency ${\omega }_{\perp }(t)$. On the left are plotted the resulting variations in the central density $\rho (t,r=0)$ when ${\omega }_{\perp }^2$ is varied according to Eq. (12) with ${\omega }_{\perp f}^2/{\omega }_{\perp i}^2=2$. The three curves correspond to $a/{\omega }_{\perp i}=0.3$ (blue curve), 0.6 (purple curve) and 1.0 (yellow curve). We note that, while for the slowest variation $\rho (t,r=0)$ varies adiabatically, oscillations appear for larger $a$, and are more pronounced the more rapid is the change in ${\omega }_{\perp }$. In the limit of a sudden change, i.e. $a/{\omega }_{\perp i}\to \infty$, their amplitude reaches a maximum, so that after the change $\rho (t,0)/\rho (t_i,0)$ oscillates between 1 and another (computable) value. On the right is shown, on a logarithmic scale, the relative amplitude of oscillations in ${\sigma }^2$ after the change, as a function of the inverse rate ${\omega }_{\perp i}/a$. There, the three curves correspond to ${\omega }_{\perp f}^2/{\omega }_{\perp i}^2=10$ (blue), 2 (purple) and 1.1 (yellow). We note that, for ${\omega }_{\perp i}/a\gtrsim 2$, these curves become proportional to $\exp (-\pi {\omega }_{\perp i}/a)$, whereas for ${\omega }_{\perp i}/a\to 0$ they approach values that can be calculated assuming a sudden change in ${\omega }_{\perp }^2$.

Figure 3

The response of the condensate to a temporal change in ${\omega }_{\perp }$ in which the trap is switched off, i.e. the final value of ${\omega }_{\perp }$ is zero. As in Fig. 2, the variation in ${\omega }_{\perp }^2$ follows (12). Plotted are the resulting variations in the central density ${\rho }_0$. The black dashed curve corresponds to the limiting case of a sudden switch-off, i.e., $a/{\omega }_{\perp i}\to \infty$. The three solid curves correspond to three different values of the rate $a$: $a/{\omega }_{\perp i}=0.3$ (blue), 0.6 (purple) and 1.2 (yellow).

Figure 4

The density-density correlation function $G_{2,k}$ in a thermal BEC, before any stimulation of phonons. This function is time-independent and, when considered as a function of $k\xi$, depends only on $T/m{c}^2$. As explained in the text, it does not depend on the thickness of the condensate. Here, the various curves correspond to four different values of the temperature: $T/m{c}^2=0$ (thick black), 0.25 (blue), $1/\sqrt{3}$ (purple) and 1 (yellow). The purple curve marks the transition between having a minimum at $k=0$ for $T/m{c}^2<1/\sqrt{3}$ and a local maximum for $T/m{c}^2>1/\sqrt{3}$. We note that all curves approach $T/m{c}^2$ in the limit $k\xi \to 0$, and they approach ${(u_k+v_k)}^2$ when ${\omega }_k/T\gg 1$.

Figure 5

The response of the in situ correlation function of Eq. (45) to a smooth variation in $c^2$ described by (46). On the left are shown three curves showing the evolution of $G_2$ with time, for a fixed wave vector $k{\xi }_f=1$, for a relative change $c_i^2/c_f^2=1/2$, and at zero temperature. The three curves differ only in the rate of change, namely $a/{\omega }_{ki}=0.3$ (blue), 0.6 (purple) and 1.0 (yellow). We see that, just as with the response of the condensate width in Fig. 2, the final $G_2$ can either settle down on a single value or oscillate around that value, depending here on the ratio $a/{\omega }_{ki}$. On the right, we have fixed $m{c}_f^{2}t=5$ and allowed the wave vector $k$ to vary. The thick black curve gives the zero-temperature adiabatic solution, which is equal to the nonseparability threshold; see condition (44). The solid blue and purple curves correspond to the rate $a={\omega }_k(k{\xi }_f=3)$, with $T_i/m{c}_i^2=0$ (blue) and $T_i/m{c}_i^2=1$ (purple). The dotted curve is the adiabatic solution at $T_i/m{c}_i^2=1$. To facilitate the reading, we have used dashed curves to show the maximum and minimum values reached as $G_2$ oscillates in time. For the purple curve, the high initial temperature has destroyed the nonseparability for long wavelengths, but it is preserved for $k{\xi }_f\gtrsim 1.5$. In both cases, the nonseparability is much less visible for $k{\xi }_f>3$ as the evolution there becomes adiabatic, so the number of excited phonon pairs drops rapidly to zero.

Figure 6

Nonseparability of phonon pairs due to a smooth change in $c^2$. In the left panel, we plot the minimum value of $G_2$ after the change, i.e. the lower envelope plotted on the right of Fig. 5. The thick black curve shows again the final profile of ${(u_k+v_k)}^2$. The change in $c^2$ is of the same sort considered in Fig. 5, except that here we use three different values of the rate $a$, which is set equal to the initial phonon frequency of three different wave vectors: $k{\xi }_f=0.3$ (blue), 1 (purple) and 3 (yellow). The solid and dashed curves correspond to initial temperatures of $T_i/m{c}_i^2=0$ and 1, respectively. On the right panel are shown the corresponding plots of ${\mathrm{\Delta }}_k\equiv n_k-|c_k|$, which (the state being Gaussian) is less than zero iff the two-mode phonon state $(k,-k)$ is nonseparable; see condition (32). From both plots, we see that a higher rate of change tends to increase the degree of nonseparability, while a higher initial temperature decreases and, as for the blue and purple curves, can even destroy it.

Figure 7

In situ observables for the resonant mode in an oscillating condensate as function of the number $N$ of oscillations. Here, we take experimental parameters similar to those used in [11]: ${\omega }_{\perp f}/{\omega }_{\perp i}=\sqrt{2}$, ${\omega }_{\perp i}/m{c}^2=1.5$, $\delta (c^2)/c^2=0.5$, but we work with a vanishing initial temperature. On the left, in blue, is shown $G_{2,k}(t)$, while its upper and lower envelopes are shown in dashed red. The thick gray line shows ${(u_k+v_k)}^2$, which is smeared due to the fact that it oscillates in response to the oscillations in $c^2$. On the right are shown ${|{\beta }_k(t)|}^2$ (in blue) and, for $T=0$, ${\mathrm{\Delta }}_k(t)$ (in red). Note, in the left plot, that the fraction of an oscillation for which $G_{2,k}(t)<{(u_k+v_k)}^2$ gets smaller with increasing $n_k$, since this causes the amplitude to increase but does not affect the minimum, which saturates at zero. When measuring $G_{2,k}(t)$, then, the nonseparability of the state becomes less visible with time.

Figure 8

Mean occupation number near the resonant mode due to natural oscillations of the condensate, as a function of $k\xi$ where $\xi$ is here the upper limit of the healing length during the modulation. We use the same parameters as in Fig. 7. The left plot shows ${|{\beta }_k|}^2$, or equivalently, $n_k$ at zero temperature; the right plot shows the corresponding $n_k$ at temperature $T/{\omega }_{\perp }=2.8$, as in [11]. The blue curves correspond to the number of oscillations $N=60$, in accordance with [11], while the red and yellow curves correspond to $N=30$ and $N=15$, respectively. The horizontal black line corresponds to a theoretical estimate for the maximum possible ${|{\beta }_k|}^2$ before the resonant mode begins to interact nonlinearly with other modes.

Figure 9

Opening the trap, at zero temperature. The condensate and phonons are initially in their ground state, and the condensate expands in response to a reduction in ${\omega }_{\perp }$, which vanishes asymptotically. The final atomic state is shown as a function of $k\xi$, where $\xi$ is the healing length in the condensate before the opening of the trap. On the left is plotted the final occupation number of atoms, while on the right is plotted the final nonseparability parameter ${\mathrm{\Delta }}_k^a$ of Eq. (51). The black dashed curve on the left plot shows ${|v_k|}^2$, the initial number of atoms with wavenumber $k$, while on the right it corresponds to ${|v_k|}^2-|u_k{v}_k|$ which is the value of ${\mathrm{\Delta }}_k^a$ characterizing the atoms in the depletion. The solid curves correspond to a sudden change in ${\omega }_{\perp }$, so that the cloud expands completely freely. The various colors correspond to different values of $n_1{a}_s$: 10 (blue), 3 (purple), and 0.7 (yellow). (These are equivalent to initial values of ${\omega }_{\perp }/m{c}^2=0.3$, 0.6 and 1.4, respectively.) The large dots on these curves show where the initial phonon frequency ${\omega }_{ki}$ is equal to ${\omega }_{\perp }$, and mark the boundary between phonons which see the expansion as sudden and are highly excited (${\omega }_{ki}\ll {\omega }_{\perp }$) and phonons which see the expansion as adiabatic and are hardly excited at all (${\omega }_{ki}\gg {\omega }_{\perp }$). The dotted curves, by contrast, correspond to a gradual opening of the trap, with the condensate having initially the same thickness as the solid yellow curve. For these, ${\omega }_{\perp }$ varies like a hyperbolic tangent (as was also used in Fig. 3), with the rate $a/{\omega }_{\perp i}$ taking the values 0.85 (right curve) and 0.3 (left curve).

Figure 10

Opening the trap after a smooth change in $c^2$, at zero temperature. We begin with the situation shown in Fig. 5, followed by a variation of ${\omega }_{\perp }$ that goes to zero asymptotically. The final atomic state is shown as a function of $k\xi$, where $\xi$ is the healing length in the condensate after the initial DCE process but before the opening of the trap. In the upper row, ${\omega }_{\perp }$ is switched off suddenly from an initial value of ${\omega }_{\perp }/m{c}^2=1$, after a time delay of $m{c}^2\mathrm{\Delta }t=5$ (the same moment at which the in situ state in Fig. 5 is drawn). The thick black curves plot the phonon state after the initial DCE, while the solid blue curves show the state of the atoms after the expansion and the dotted curves plot the upper and lower envelopes of the oscillations. In the lower row, only the envelopes are shown, and we now include thick dashed curves to show the state that would result from the cloud expansion alone. The blue curves correspond to the same sudden switch-off of ${\omega }_{\perp }$ as used in the upper row, while the green and red curves correspond instead to a gradual decrease of ${\omega }_{\perp }$ of the kind used in Figs. 3 and 9. The rates used for the switch-off are $a/{\omega }_{\perp }=0.6$ (green) and 0.2 (red). It is clear that slowing the rate of expansion greatly reduces its effect, and in this way we can bring the final state of the atoms closer to the state of the phonons that existed before the expansion.

Figure 11

Relative difference between the numerical density profile and the Gaussian approximation. The chemical potential $\mu$ takes the values $1.01{\omega }_{\perp }$, $1.1{\omega }_{\perp }$, $1.42{\omega }_{\perp }$, and $2.475{\omega }_{\perp }$ from blue to green, corresponding to $n_1{a}_s\approx 0.01$, 0.05, 0.25, and 1.25, respectively.

Figure 12

Plot of the quantity $({\mu }^2/{\omega }_{\perp }^2-1)/(4n_1{a}_s)$, which is identically equal to 1 in the Gaussian approximation. The distance from unity is linear in $n_1{a}_s$ when the latter becomes much smaller than 1.

Figure 13

Plots of the ratio of the frequency $\omega$ numerically computed over the value ${\omega }_G$ obtained using the approach of Sec. 3a. The background solutions are the same as in Fig 11.

Figure 14

Left: Plot of $\omega$ versus $k$ for the first four branches of the dispersion relation on the cylindrically-symmetric, node-less stationary solution with $\mu =1.42{\omega }_{\perp }$, corresponding to $n_1{a}_s\approx 0.25$. The wave vector $k$ and angular frequency $\mathrm{\Omega }$ are adimensionalized using the healing length ${\xi }_G$ and effective one-dimensional sound speed $c_G$ derived using the Gaussian approximation. Right: Angular frequency at $k=0$ for the second to fourth branches as a function of $n_1{a}_s$, in logarithmic scale.

Figure 15

The final state of the atoms after TOF when the initial state is a thermal bath of phonons. As in the lower row of Fig. 10, the solid black curves show the state before the expansion (but after the initial DCE), $\xi$ is the healing length before the expansion, the colored dashed curves show the state that would result from the expansion alone, and the colored regions show the envelopes within which $n_k^a$ and ${\mathrm{\Delta }}_k^a$ will oscillate. The initial temperature is taken to be $T_i/m{c}_i^2=0.5$, while the rates of opening the trap are the same as in Fig. 10: $a/{\omega }_{\perp }=\infty$ (i.e. sudden switch-off, blue), 0.6 (green) and 0.2 (red). Interestingly, for $k\to 0$, the distribution becomes nonseparable with a value of ${\mathrm{\Delta }}_k^a$ that is independent of $a/{\omega }_{\perp }$.

Figure 16

Final state of atoms after a modulation of $c^2$, at non-zero temperature. We consider the situation described in Ref. [11]. The parameters used are ${\omega }_{\perp f}/{\omega }_{\perp i}=\sqrt{2}$, ${\omega }_{\perp i}/m{c}^2=1.5$, $\delta (c^2)/c^2=0.5$ and $T/{\omega }_{\perp }=2.8$. ${\omega }_{\perp }$ is immediately switched off after an integer number of oscillations, after which the condensate expands freely. On the left is shown the final occupation number of atoms, while on the right is shown the nonseparability parameter ${\mathrm{\Delta }}_k^a$. The various curves correspond to a different number of oscillations of the condensate before the trap is opened: $N=4$ (blue curves), 6 (red curves) and 10 (yellow curves). Higher values of $N$ (namely 60, 30 and 15) were considered in Fig. 8.

Figure 17

Final two-atom correlation function, $g_2(k,-k)$. On the left, the situation is exactly the same as in Fig. 16, whereas on the right the temperature has been divided by 2, i.e. it has $T/{\omega }_{\perp }=1.4$. As in Fig. 16, the different curves correspond to different total numbers of oscillations of the condensate: 4 (blue), 6 (red) and 10 (yellow). Note that $g_2(k,-k)$ varies nonmonotonically with $N$, particularly at resonance. Thus, just as with the in situ density-density correlation function (see Fig. 7 and its caption), nonseparability becomes less visible with larger $n_k^a$.