Filtering single atoms from Rydberg blockaded mesoscopic ensembles
We propose an efficient method to filter out single atoms from trapped ensembles with unknown number of atoms. The method employs stimulated adiabatic passage to reversibly transfer a single atom to the Rydberg state which blocks subsequent Rydberg excitation of all the other atoms within the ensemble. This triggers the excitation of Rydberg blockaded atoms to short lived intermediate states and their subsequent decay to untrapped states. Using an auxiliary microwave field to carefully engineer the dissipation, we obtain a nearly deterministic single-atom source. Our method is applicable to small atomic ensembles in individual microtraps and in lattice arrays.
pacs:32.80.Ee, 32.80.Qk, 37.10.Jk, 37.10.Gh
Introduction.— Deterministic preparation of single trapped atoms, or arrays thereof, is an important task required for many potentially interesting applications, including quantum computation and simulation Jaksch1999 (); Jaksch2000 (); Schrader2004 (); Schneider2012 (); OLsimulators (); Bloch2008 (); Weimer2010 (). Optical lattice experiments can realize Mott insulators with single atom per lattice site RevOL () with low particle or hole defect probability Bakr2010 (); Sherson2010 (). Achieving single-atom occupation of spatially separated microtraps with high fidelity is more challenging. Several techniques have been explored to reduce the trap occupation number fluctuations Mohring2005 (); Gruenzweig2010 (); Bakr2011 (); Nikolopoulos2010 (); Rabl2003 (); Sherson2012 (); delCampo2008 ()
Here we propose a method to filter out single atoms from trapped ensembles with unknown number of atoms. Our scheme relies on the strong, long-range interaction between atoms in high-lying Rydberg states RydAtoms (); rydrev () which blocks resonant optical excitation of more than one Rydberg atom within a certain interatomic distance Jaksch2000 (); Lukin2001 (); Vogt2006 (); Tong2004 (); Singer2004 (); Heidemann2007 (); NatPRLGrangier (); NatPRLSaffman (); Dudin2012 (); Schauss2012 (); Robicheaux2005 (); Stanojevic (); Petrosyan (); Ates2013 (); Petrosyan2014 (). We use stimulated adiabatic passage technique (STIRAP) stirap () to transfer any one atom of the ensemble from the ground state to the Rydberg state by a two-photon transition via an intermediate excited state Moeller2008 (); Petrosyan2013 (); Beterov2011 (); schempp10 (). This atom then strongly suppresses Rydberg excitation of the other atoms due to the interaction-induced level shifts. The Rydberg-blockaded atoms now turn into strongly-driven two-level atoms which rapidly decay from the (intermediate) excited state back to the initial trapped ground state Petrosyan2013 () as well as to other untrapped states. After several excitation and decay cycles, all of the Rydberg-blockaded atoms are removed from the trap. The reverse adiabatic transfer of the remaining Rydberg atom back to the initial ground state completes the preparation of a single trapped atom. We achieve high efficiency and reliability of the single-atom source by carefully engineering the atomic excitation and dissipation. The former is controlled by the time-dependent amplitude and detuning of the laser field, while the latter can be accomplished by employing an auxiliary microwave field to open appropriate decay channels to the untrapped states.
Stimulated adiabatic passage with a single-atom.— For what follows, it will be helpful to recall the essence of adiabatic transfer (STIRAP) of population in an isolated three-level atom using a pair of coherent laser pulses stirap (). A laser field with Rabi frequency couples the stable ground state to an unstable (decaying) excited state , which in turn is coupled to another stable state by the second laser field of Rabi frequency [Fig. 1(a)]. In the frame rotating with the frequencies of the laser fields, the atom-field interaction Hamiltonian reads
where is the detuning of the intermediate state and we assume two-photon resonance between and . The eigenvalues of are and and the corresponding eigenstates are given by
where and . The zero-energy “dark” state does not contain the rapidly decaying state , which, however, contributes to the energy-shifted “bright” states making them unstable against spontaneous decay. By adiabatically changing the dark state superposition, the STIRAP process can completely transfer the population between the two stable states and without populating the unstable state . To this end, with the atom initially in state , one first applies the field (), resulting in . This is then followed by switching on the field, leading to , which results in when . For sufficiently smooth switching, , the atom adiabatically follows the dark state , and the bright states , and thereby , are never populated. The reverse process, at the end of which , can transfer the atom in state back to state .
The multiatom system.— Consider now an ensemble of atoms initially in the ground state confined within a small trap. All the atoms uniformly interact with two laser fields on the transitions and with Rabi frequencies and , as shown in Fig. 1(a). For each atom , the interaction Hamiltonian reads
where are the atomic operators. The intermediate excited state , possibly detuned by , decays with the rate to the trapped ground state , and with the rate to another untrapped lower state which is decoupled from the driving lasers. For completeness, we also take into account the typically much smaller decay rate of the highly excited Rydberg state and its possible dephasing (with respect to the other states) due to nonradiative collisions, inhomogeneous trapping potential, external field noise, laser phase fluctuations or Doppler shifts. The corresponding Lindblad generators for the decay and dephasing processes are , , , and , where .
The long-range atom-atom interactions are described by the Hamiltonian
where is the van der Waals potential between pairs of atoms in the Rydberg state separated by distance . When the interaction-induced level shifts of states exceed the two-photon excitation linewidth of the Rydberg state (), double (multiple) Rydberg excitations are blocked. We can then define the blockade distance via leading to . To achieve nearly complete blockade of more than one Rydberg excitation, we assume that all the atoms are initially trapped within a volume of linear dimension corresponding to .
Numerical simulations.— We simulate the dissipative dynamics of the system of atoms using the quantum stochastic (Monte Carlo) wavefunctions qjumps (), which allows us to treat nontrivial numbers of strongly-interacting atoms exactly, without truncating the correspondingly large Hilbert space. In such simulation, the state of the system evolves according to the Schrödinger equation with an effective Hamiltonian
is the usual (Hermitian) Hamiltonian, while
is the non-Hermitian part which does not preserve the norm of . The evolution is interrupted by random quantum jumps of individual atoms with probabilities determined by the corresponding weights . In a single quantum trajectory, the normalized wavefunction of the system at any time is given by . The expectation value of any observable of the system is then obtained by averaging over many, , independently simulated trajectories, , while the density operator is given by .
Starting from all the atoms in the trapped ground state, , we assume that during the evolution the atoms that decay to state decouple from the laser field and are eventually lost from the trap (which can be accelerated by a pushing laser). The probabilities for , out of the initial , atoms surviving in the trap are then given by the projectors , , etc. The mean number of trapped atoms is then . Typically, the initial number of trapped atoms is uncertain, with the Poisson distribution
around some mean . Then, the probabilities for atoms surviving in the trap are given by .
We have performed numerical simulations for various initial number of atoms in the trap, results of which are shown in Figs. 2-4. In all simulations, we first apply the field coupling states and with the Rabi frequency which stays constant throughout the process, and then send a smooth pulse driving transition with the Rabi frequency whose peak value is much larger than . We thus expect that a single atom in the trap will be adiabatically transferred from the ground state to the Rydberg state and block Rydberg excitations of all the other atoms initiating thereby their optical pumping through the intermediate excited state to the decoupled and untrapped state(s) . At the end of the process, when is reduced to zero, the remaining Rydberg-state atom will be transferred back to the trapped ground state .
Results.— Consider first the resonant excitation, , of the atoms with the and fields shown in the upper panel of Fig. 2. The branching ratio for the decay of the atoms from the intermediate excited state to the untrapped state and back to is large , as per Fig. 1(a), while we neglect for the moment the dephasing of the Rydberg state, . We find that even before the pulse attains its maximum, most of the atoms are optically pumped into the untrapped state(s) , while the mean number of atoms in the trap drops below unity: and , while . The reason for this unexpected and unfortunate behavior is that each quantum jump of a blocked atom from state to either or projects a single atoms to the Rydberg state Petrosyan2013 () whose overlap with the dark state is smaller than unity, unless . Since for the Rydberg atom is not in the dark eigenstate, but has large overlap with the bright eigenstates as well, , it is coupled to the state from where it can decay to or . In a closed three-level system Petrosyan2013 (), the decay to would simply restart the excitation process of this or any other atom by the fields and . In the present case, however, the decay of all the atoms but one to the decoupled state may leave the last atom in the Rydberg state before the pulse reaches its peak value . The probability of loosing this last remaining atom can thus be estimated as
which, with and is about , consistent with and in Fig. 2.
Including now in the simulations realistic dephasing, , amounts to additional coherence relaxation between the long-lived states and with the rate . As is well known stirap (), the STIRAP in a three-level atom is very sensitive to such dephasing, since the decoherence of the dark-state superposition leads to additional population of the bright states Petrosyan2013 (), which in the present system can decay to the untrapped state(s) . Hence, in the inset of Fig. 2 we observe further decrease of the survival probability for a single trapped atom.
To increase the single atom survival probability , we should therefore postpone the decay of atoms from state to until is much larger than . We thus take a finite detuning of intermediate level and switch-on slightly faster. The results of simulations are shown in Fig. 3, were we indeed notice significant improvement of the single-atom filtering, or with additional dephasing . These results are obtained after some optimization of both and , but for the given atomic level scheme further substantial improvements do not seem possible.
To achieve nearly perfect filtering, we now consider the closed three-level system of Fig. 1(b), where state decays spontaneously only to the trapped state , while we can engineer an additional decay channel to the untrapped states . To this end, we use a microwave field which weakly couples to an auxiliary state with the Rabi frequency that is small compared to the total decay rate of . Upon adiabatic elimination of , we obtain a small additional decay on the transition with rate , and a decay to the untrapped states with rate which can be controlled via the microwave field intensity (the numerical prefactors are related to the branching ratio of the decay of state to and , ). Figure 4 shows the results of our simulations demonstrating dramatically improved single atom filtering, [ and , ], at the expense of a longer duration of the process. Even sizable dephasing of the Rydberg state with respect to the ground state does not significantly reduce the efficiency of the single-atom filtering, , which is due to the robustness of adiabatic passage in a Rydberg-blockaded ensemble composed of atoms with nearly-closed transitions Petrosyan2013 ().
We finally note that in obtaining the averaged probabilities of the final number of atoms, we assumed the initial Poisson distribution of the atom number in the trap with the mean , for which and , and performed simulations for . The initial 2% population of the trap not included in our simulation can only increase the final single atom probability. Our method should work even better for larger initial number of atoms in the trap, which is, however, prohibitively difficult to simulate numerically.
Summary.— To conclude, we have proposed and analyzed a viable scheme to filter out single atoms from mesoscopic ensembles containing unknown number of initially trapped atoms. We have shown that the combination of two-photon adiabatic transfer of the atoms to the strongly interacting Rydberg state and carefully engineered decay of atoms from the intermediate excited state can lead to nearly-deterministic production of single trapped atoms in individual microtraps or optical lattice potentials, which is an important prerequisite for quantum computation with atoms serving as qubits or quantum simulations of many-body spin systems. Moreover, our studies represent an important and interesting new facet of the very active field of dissipative generation of states and processes in open many-body systems Diehl2008 (); Verstraete2009 ().
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