Pumped-Up SU(1,1) Interferometry

Although SU(1,1) interferometry achieves Heisenberg-limited sensitivities, it suffers from one major drawback: Only those particles outcoupled from the pump mode contribute to the phase measurement. Since the number of particles outcoupled to these “side modes” is typically small, this limits the interferometer’s absolute sensitivity. We propose an alternative “pumped-up” approach where all the input particles participate in the phase measurement and show how this can be implemented in spinor Bose-Einstein condensates and hybrid atom-light systems—both of which have experimentally realized SU(1,1) interferometry. We demonstrate that pumped-up schemes are capable of surpassing the shot-noise limit with respect to the total number of input particles and are never worse than conventional SU(1,1) interferometry. Finally, we show that pumped-up schemes continue to excel—both absolutely and in comparison to conventional SU(1,1) interferometry—in the presence of particle losses, poor particle-resolution detection, and noise on the relative phase difference between the two side modes. Pumped-up SU(1,1) interferometry therefore pushes the advantages of conventional SU(1,1) interferometry into the regime of high absolute sensitivity, which is a necessary condition for useful quantum-enhanced devices.

Figure 1

(a) A conventional SU(1,1) interferometer, constructed with two active nonlinear beam splitters ${\widehat{U}}_{\mathrm{PA}}(r)$. (b) Pumped-up SU(1,1) interferometry with three modes of a spinor BEC. Initially, all atoms are in the $m_F=0$ pump mode, assumed to be a coherent state $|{\alpha }_0⟩$ with $|{\alpha }_0{|}^2=\overline{N}$. The active beam splitter ${\widehat{U}}_{\mathrm{SMD}}$ is achieved via spin-mixing collisions between three hyperfine levels [see (i) and Eq. (2)], whereas the pump is mixed with the two side modes using a tritter ${\widehat{U}}_{\mathrm{tr}}(\theta )$, engineered with coherent radio frequency pulses [see (ii) and Eq. (5)]. (c) Pumped-up SU(1,1) interferometry with the four modes of a hybrid atom-light system. The initial pump modes are coherent states, the active beam splitter ${\widehat{U}}_{\mathrm{R}}(r)$ is realized by FWM engineered with a Raman process [see (iii) and Eq. (3)], and pump enhancement is achieved with atomic and optical beam splitters that separately mix the atomic and photonic modes [see (iv) and (v), respectively].

Figure 2

Comparison of pumped-up and conventional SU(1,1) interferometry, engineered within (a) a spinor BEC and (b) a hybrid atom-light system (with $n_f=1$). The total particle number is $\overline{N}=10^4$. Sensitivities are plotted in (i), while optimal tritter (or beam splitter) angles ${\theta }_{\mathrm{opt}}$ for pumped-up interferometry are shown in (ii). For our pumped-up schemes, $\mathrm{\Delta }{\varphi }_{\min }=1/\sqrt{\mathcal{F}({\theta }_{\mathrm{opt}})}$ is the QCRB, and $(\mathrm{\Delta }{\varphi }_N{)}^2={\min }_{\theta ,\varphi }\mathrm{Var}({\widehat{N}}_s)/|\partial ⟨{\widehat{N}}_s⟩/\partial \varphi {|}^2$ gives the phase sensitivity for a number-sum measurement of the two side modes at the output; these are plotted for ${\vartheta }_{\mathrm{opt}}$. $\mathrm{\Delta }{\varphi }_{\mathrm{SU}(1,1)}=1/\sqrt{\mathcal{F}(0)}$ is the QCRB for conventional SU(1,1) interferometry, only saturated by a number-sum measurement of the side modes within the undepleted pump regime. Solid lines are analytic curves obtained in the undepleted pump regime (accurate to all orders of $\overline{N}$—see [31] for exact expressions), whereas markers are truncated Wigner simulations which include the effects of pump depletion [45]. The four vertical lines indicate the degree of squeezing associated with four values of $⟨{\widehat{N}}_s⟩$; these mark experimentally accessible regimes ranging from currently achievable (3 dB) to extremely challenging (20 dB).

Figure 3

Relative sensitivities of pumped-up SU(1,1) interferometry compared with conventional SU(1,1) interferometry in (a) a spinor BEC setup [top panels] and (b) a hybrid atom-light system (with $n_f=1$) [bottom panels]. The total particle number is $\overline{N}=10^4$. All values plotted are at optimal $\varphi$ and, for pumped-up schemes, optimal angle ${\theta }_{\mathrm{opt}}$ and phase ${\vartheta }_{\mathrm{opt}}$. These show the dependence on (left) the fraction of particles lost due to (a) two-body and (b) one-body losses, obtained via truncated Wigner simulations; (middle) imperfect particle detection with number resolution $\mathrm{\Delta }n$, obtained from semianalytic calculations [31]; and (right) Gaussian phase-difference noise of variance ${\sigma }_{\phi }^2$, obtained from analytic calculations with $\varphi$ optimized numerically. Pumped-up SU(1,1) interferometry is superior to conventional SU(1,1) interferometry for those points or curves outside the shaded region. The side-mode populations ${\mathcal{N}}_s=10$, 50, and 500 correspond to approximately 13.4, 20, and 30 dB of squeezing, respectively. Absolute sensitivities $\mathrm{\Delta }{\varphi }_N$ under losses are plotted in Supplemental Material [31].