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We have studied the decay pattern of the $J^{\pi} =1_{4742~\mathrm{keV}}^{-}$ state of $^{88}$Sr to probe its quadrupole-octupolecoupled two-phonon structure. In particular, a unique fingerprint to prove thetwo-phonon nature is the $E2$ decay strength of the $1_{4742~\mathrm{keV}}^{-}\to 3_{1}^{-}$ transition into the one-octupole-phonon state. $\gamma$-rayspectroscopy was performed after the $\beta$-decay of $^{88}$Rb to obtain thenecessary sensitivity for this weak-intensity decay branch. Sufficient amountsof $^{88}$Rb (T$_{1/2}$ = 17.8 min) were produced by neutron activation ofnatural Rb in the TR IGA Mark II reactor. The results show that the $B(E2)$value of the $1_{4742~\mathrm{keV}}^{-} \to 3_{1}^{-}$ transition is equal tothe $B(E2)$ of the $2_{1}^{+} \to 0_{1}^{+}$ transition, directly demonstratingthe quadrupole-octupole coupled two-phonon nature of the $1^-$ state. Acomparison of the results with energy-density functional plusquasiparticle-phonon model calculations shows rema rkable agreement,corroborating this assignment.

Direct demonstration of the two-phonon structure of the $J^{\pi} = 1_{4742~\mathrm{keV}}^{-}$ state of $^{88}$Sr