Quantum free-fall motion and quantum violation of the weak equivalence principle

The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approaches to the problem. Here, we tackle the problem anew through the time of arrival (TOA) operator approach. This is done by constructing the TOA operator for a nonrelativistic and structureless particle that is projected upward in a uniform gravitational field with an intended arrival point below the classical turning point. The TOA operator is constructed under the constraint that the inertial and gravitational masses are equivalent and that Galilean invariance is preserved. These constraints are implemented by Weyl quantization of the corresponding classical TOA function for the projectile. The expectation value of the TOA operator is explicitly shown to be equal to the classical time of arrival plus mass-dependent quantum correction terms, implying incompatibility of the WEP with quantum mechanics. The full extent of the violation of the WEP is shown through the mass dependence of TOA distribution for the projectile.

Figure 1

Time evolution of the nodal (top) and non-nodal (bottom) eigenfunctions of the TOA operator with eigenvalues 0.00508 a.u. and 0.00517 a.u., respectively, for the parameters $\mu =g=1$. All quantities are in atomic units.

Figure 2

Quantum time of arrival at the origin for a single atom described as a Gaussian wave packet where the classical turning point is below the arrival point for the parameters $\mu =g=\hslash =1$, initial velocity $v_o=2$, and ${\sigma }^2=0.1$ for varying initial position $q_o$.

Figure 3

Magnitude of the leading quantum correction term to the classical TOA for a single atom described as a Gaussian wave packet with increasing initial velocity for the parameters $\hslash =g=1$, initial position $q_o=-5$, arrival point at the origin $q=0$, and ${\sigma }^2=0.1$.

Figure 4

Magnitude of the leading quantum correction term to the classical TOA for a single atom described as a Gaussian wave packet with increasing initial velocity for the parameters $\hslash =g=1$, initial position $q_o=-5$, arrival point at the origin $q=0$, and ${\sigma }^2=0.1$ for $m_i\ne m_g$.

Figure 5

Magnitude of the leading quantum correction term to the classical TOA for a single atom described as a Gaussian wave packet with increasing initial velocity for the parameters $\mu =g=\hslash =1$, and arrival point at the origin $q=0$: (top) the spread is kept constant at ${\sigma }^2=0.1$ with varying initial position; (bottom) the initial position is kept constant at $q_o=-5$ with varying spread of the wave packet.

Figure 6

The time-evolved position density distribution of a single atom described as a Gaussian wave packet with initial velocity $v_o=20$, initial position $q_o=-3\left[{\text{J}}^{-1}\right]$, and ${\sigma }^2=0.1\left[{\text{J}}^{-2}\right]$ for the parameters $\mu =g=1\left[\text{J}\right]$ (top) with its corresponding time of arrival distribution at the origin (bottom).

Figure 7

Time of arrival distribution of a single atom described as a Gaussian wave packet for the parameters $g=\hslash =1$ with initial initial velocity $v_o=20$, initial position $q_o=-5$, arrival point at the origin $q=0$, and ${\sigma }^2=0.1$

Figure 8

Time of arrival distribution of a single atom described as a Gaussian wave packet for the parameters $\hslash =g=1$ with initial velocity $v_o=20$, initial position $q_o=-5$, arrival point at the origin $q=0$, and ${\sigma }^2=0.1$ for $m_i\ne m_g$.

Figure 9

Time of arrival distribution of a single atom described as a Gaussian wave packet for the parameters $\mu =g=\hslash =1$ with ${\sigma }^2=0.1$ and $q_o=-5$ (top), $v_o=20$ and $q_o=-5$ (middle), and $v_0=20$ and ${\sigma }^2=0.1$ (bottom).