Non-Markovian polaron dynamics in a trapped Bose-Einstein condensate

We study the dynamics of an impurity embedded in a trapped Bose-Einstein condensate, i.e., the Bose polaron problem. This problem is treated by recalling open quantum systems techniques: the impurity corresponds to a particle performing quantum Brownian motion, while the excitation modes of the gas play the role of the environment. It is crucial that the model considers a parabolic trapping potential to resemble the experimental conditions. Thus, we detail here how the formal derivation changes due to the gas trap, in comparison to the homogeneous gas. More importantly, we elucidate all aspects in which the gas trap plays a relevant role, with an emphasis on the enhancement of the non-Markovian character of the dynamics. We first find that the presence of a gas trap leads to a new form of the bath-impurity coupling constant and a larger degree in the super-Ohmicity of the spectral density. We then solve the quantum Langevin equation to derive the position and momentum variances of the impurity, where the former is a measurable quantity. For the particular case of an untrapped impurity, the asymptotic behavior of this quantity is found to be motion superdiffusive. When the impurity is trapped, we find position squeezing, casting the system suitable for implementing quantum metrology and sensing protocols. We detail how both superdiffusion and squeezing can be enhanced or inhibited by tuning the Bose-Einstein condensate trap frequency. Compared to the homogeneous gas case, the form of the bath-impurity coupling constant changes, and this is manifested as a different dependence of the system dynamics on the past history. To quantify this, we introduce several techniques to compare the different amount of memory effects arising in the homogeneous and inhomogeneous gas. We find that it is higher in the second case. This analysis paves the way to the study of non-Markovianity in ultracold gases, and the possibility to exploit such a property in the realization of new quantum devices.

Figure 1

Time dependence of the function $G_2$, defined through its Laplace transform in Eq. (36). The thick lines represent the numerical calculation with the Zakian algorithm, while the corresponding continuous thin ones refer to the expression in Eq. (47), valid in the long-time limit.

Figure 2

Superdiffusion coefficient in Eq. (52) as a function of the interaction strength for different values of the gas trap frequency. We present the results for an impurity of Yb embedded in a Rb gas of $N=50000$ atoms with coupling strength $g_{\mathrm{B}}={10}^{-38}\text{J}\text{m}$. In this context the units of frequency are ${\omega }_{\mathrm{c}}=\frac{m_{\mathrm{I}}{g}_{\mathrm{B}}^2}{{\hslash }^3}$, while the units of the length are $l_{\mathrm{c}}=\frac{{\hslash }^2}{m_{\mathrm{I}}{g}_{\mathrm{B}}}$.

Figure 3

Time dependence of the function $G_1$ (top) and $G_2$ (bottom), defined through the Laplace transforms in Eqs. (35) and (36), respectively. The plots refer to an impurity of Yb in a trap with a frequency $\mathrm{\Omega }=2\pi \times 200$ Hz, embedded in a Rb gas of $N=5000$ atoms with trap frequency ${\omega }_{\mathrm{B}}=2\pi \times 800$ Hz and coupling strength $g_{\mathrm{B}}=0.5\times {10}^{-37}$ J m.

Figure 4

Temperature dependence of the ratio ${\delta }_p/{\delta }_p$ between the variances introduced in Eq. (57). The plot refers to an impurity of Yb in a trap with a frequency $\mathrm{\Omega }=2\pi \times 50$ Hz, embedded in a Rb gas of $N=5000$ atoms with trap frequency ${\omega }_{\mathrm{B}}=2\pi \times 500$ Hz and coupling strength $g_{\mathrm{B}}=0.6\times {10}^{-38}$ J m.

Figure 5

Temperature dependence of the position variance introduced in Eq. (57), for different values of the coupling strength. The plot refers to an impurity of Yb in a trap with a frequency $\mathrm{\Omega }=2\pi \times 200$ Hz, embedded in a Rb gas of $N=5000$ atoms with trap frequency ${\omega }_{\mathrm{B}}=2\pi \times 800$ Hz and coupling strength $g_{\mathrm{B}}=0.5\times {10}^{-37}$ J m. The red dot-dashed line represents the function $\sqrt{2T}$, related to the equipartition theorem.

Figure 6

Position variance introduced in Eq. (57) as a function of the coupling strength, for different values of the gas trap frequency, in the low temperature regime. The plot refers to an impurity of Yb in a trap with a frequency $\mathrm{\Omega }=2\pi \times 200$ Hz, embedded in a Rb gas of $N=5000$ atoms with trap frequency ${\omega }_{\mathrm{B}}=2\pi \times 800$ Hz and coupling strength $g_{\mathrm{B}}=0.5\times {10}^{-37}$ J m.

Figure 7

Position variance in Eq. (57) as a function of the gas trap frequency at several different values of the temperature. The plot refers to an impurity of Yb in a trap with a frequency $\mathrm{\Omega }=2\pi \times 200$ Hz, embedded in a Rb gas of $N=5000$ atoms with trap frequency ${\omega }_{\mathrm{B}}=2\pi \times 800$ Hz and coupling strength $g_{\mathrm{B}}=0.5\times {10}^{-37}$ J m.

Figure 8

Non-Markovianity measure in Eq. (58), as a function of the cutoff frequency, associated with a quartic SD (red-solid line) and a cubic (blue-dashed line) one, corresponding respectively to an inhomogeneous and a homogeneous gas.

Figure 9

Time dependence of $G_1$ (up) and $G_2$ (down) calculated for the quartic SD in Eq. (31) related to an inhomogeneous gas (solid red line) and a cubic one derived in [38] for a homogeneous medium (dashed blue line). The plot has been realized for $\mathrm{\Lambda }/\mathrm{\Omega }=10$ and $\gamma /\mathrm{\Omega }=7$.

Figure 10

Time dependence of the function $G_2$, defined through its Laplace transform in Eq. (36). The thick lines represent the result obtained in the inhomogeneous case, while the thin ones refer the homogeneous medium.

Figure 11

Non-Markovianity measure in Eq. (83), associated with SD in Eq. (31), as a function of the temperature for different values of the damping constant. The plot refers to $\mathrm{\Lambda }/\mathrm{\Omega }=10$.

Figure 12

Ratio between the non-Markovianity measure in Eq. (83) calculated for the SD associated with the homogeneous case (see Eq. (41) in [38]) and that in Eq. (31). The plot expresses a temperature dependence, for different values of the damping constant, and refers to $\mathrm{\Lambda }/\mathrm{\Omega }=10$.

Figure 13

Backflow of energy in Eq. (84) as a function of the cutoff frequency calculated for an inhomogeneous gas (red solid line) and a homogeneous one (blue dashed line).

Figure 14

Validity condition in Eq. (A8) for an untrapped impurity of Yb in a gas made up by $N=5000$ atoms of K with a coupling strength $g_{\mathrm{B}}=0.5\times {10}^{-37}$ J m, trapped in a harmonic potential with ${\omega }_{\mathrm{B}}=2\pi \times 800$ Hz.