Gravitational time dilation in quantum clock interferometry with entangled multi-photon states and quantum memories

Abstract

Gravitational time dilation implies that clocks held at different heights accumulate different proper times. We analyze a memory-assisted quantum clock interferometer in which a frequency-bin photonic clock is stored in two vertically separated quantum memories for a controllable duration, such that the joint state evolves in a quantum superposition of two proper times. After retrieval, the photonic modes interfere in a Hong-Ou-Mandel (HOM) interferometer, for which we derive analytic expressions for the resulting multiphoton detection statistics. Extending this HOM-based scheme from entangled photon pairs to frequency-entangled 2N-photon inputs, we show that the proper-time dependent phase is amplified by a factor N, leading to an N-times faster collapse and revival of the interference signal compared with the two-photon case. Incorporating finite memory efficiency and lifetime, we identify regimes where this modulation remains observable. For parameters compatible with demonstrated Rb and Cs memories and achievable optical frequency separations, the first collapse occurs for height differences in the order of 10-100 m with subsecond to few-second storage times, while suitable rare-earth ion and alkali memory combinations can reduce the required height to the few-metre scale. These results establish near-term laboratory conditions for observing entanglement dynamics driven by gravitational time dilation in a photonic platform.

Figures (3)

• Figure 1: Experimental proposal: a) The vertical interferometer with two quantum memories in each arm for frequency components $\Omega_{1,2}$. The entangled pair source (EPS) generates the state in Eq.\ref{['eq:HOM_N']} and each frequency component is stored in corresponding quantum memories in upper (U) and lower (L) branches. After local storage time $t_s$, the memories are read out and the resultant state is sent to the BS for the eventual parity detection. b) Quantum clock with photons and the relevant $\Lambda$-scheme for the memories.: a $2N$-photon state (Eq.\ref{['eq:HOM_N']}) from $\Omega_{1,2}$ frequency components which are resonant with memories with corresponding colours. $\Omega_i$ denotes the input frequency; $\Omega_i^C$ is the corresponding control pulse frequency for storage and retrieval and $\nu_i$ is the spin-wave frequency.
• Figure 2: Plot of Eqs. \ref{['eq:p_pl_pl']} and \ref{['eq:par_cosine']} for Rubidium ($780$ nm) and Cesium ($894$ nm) quantum memories, corresponding to $\Omega_- = 2\pi\times49$ THz and a vertical separation of $h=20$ m, as a function of the memory time $\tau_s$. The expected signal from a Mach--Zehnder interferometer using a single-photon frequency bin (i.e. N00N with N=1) Barzel2024entanglement is also shown for comparison, where the fast oscillations with $\Omega_+=\Omega_2+\Omega_1$ are also visible. The initial global phase, $\phi$, is set to 0.
• Figure 3: Log--log map of the gravitationally induced entanglement collapse time $t_{\mathrm{ent}}$ as a function of the differential optical frequency $\Delta f = \Omega_-/2\pi$ and height $h$ for $N=2$. The contour lines indicate equal collapse times from $10^{-2}\,\mathrm{s}$ to $10^{4}\,\mathrm{s}$. Different specific configurations are marked. (i) $\Delta f = 10$ GHz at $500$ m, which could be realized by using different spectral regions within the same rare-earth doped memory; (ii) and (iii) Rb -- Cs memory pairs at 75 m and 20 m; (iv) Pr$^{3+}:$Y$_2$SO$_5$ -- Rb pair at 3 m.