Quantum information processing with single photons and atomic ensembles
in microwave coplanar waveguide resonators
We show that pairs of atoms optically excited to the Rydberg states can strongly interact with each other via effective long-range dipole-dipole or van der Waals interactions mediated by their non-resonant coupling to a common microwave field mode of a superconducting coplanar waveguide cavity. These cavity mediated interactions can be employed to generate single photons and to realize in a scalable configuration a universal phase gate between pairs of single photon pulses propagating or stored in atomic ensembles in the regime of electromagnetically induced transparency.
pacs:03.67.Lx, 37.30.+i, 32.80.Ee, 42.50.Gy,
Ensembles of trapped atoms or molecules are promising systems for quantum information processing and communications QCcomp (). They can serve as convenient and robust quantum memories for photons, providing thereby an interface between static and flying qubits lukpet (), using e.g. stimulated Raman techniques, such as electromagnetically induced transparency (EIT) EITrev (). However, controlled interactions realizing universal quantum logic gates and entanglement in a deterministic and scalable way are difficult to achieve with photonic qubits propagating or stored in atomic ensembles.
A promising scheme for deterministic logic operations between stored photonic qubits was proposed in LFCDJCZ (). The proposal exploits the so-called dipole blockade mechanism, wherein strong resonant dipole-dipole interaction (DDI) between Rydberg atoms suppresses multiple excitations within a certain interaction volume. First proof-of-principle experiments have impressively demonstrated the blockade effect for related van der Waals interactions (VdWIs) between Rydberg atoms vdwblk (). The achieved blockade radius of only a few m is, however, not yet sufficient for implementing logic gates. Furthermore, the Rydberg blockade of LFCDJCZ () has two principle drawbacks. (i) In free-space, the DDI scales with interatomic distance as . The blockade gap in the Rydberg excitation spectrum of an atomic cloud is determined by the smallest DDI between pairs of atoms at opposite ends of the cloud. Yet, for closely spaced atoms, the DDI can be very large, which may lead to level crossings with other Rydberg states opening detrimental loss channels. (ii) Complete excitation blockade in an atomic ensemble requires spherical symmetry of the resonant DDI, which severely restricts the choice of suitable Rydberg states.
Here we put forward an alternative, scalable and efficient approach untainted by the above difficulties. We first show that superconducting coplanar waveguide (CPW) resonators BHWGS (); MCGKJS-SPS (), operating in the microwave regime, can mediate long-range controlled interactions between neutral atoms optically excited to the Rydberg states. By appropriate choice of the system parameters, effective resonant DDI or VdWI between pairs of atoms located near the CPW surface RDDLSZ () can be achieved. These interactions can then be employed to generate single photons and to realize a universal phase gate between pairs of single photon pulses propagating or stored in cold trapped atomic ensembles in the EIT regime.
Consider a pair of atoms and optically excited to the Rydberg states . The atoms interact non-resonantly with a certain mode of CPW cavity with frequency via transitions to adjacent Rydberg states and lying, respectively, above and below (Fig. 1). All the other cavity modes are far detuned from the atomic transition frequencies and and do not play a role. In the frame rotating with , the Hamiltonian is given by
where is the transition operator of the th atom, and are the corresponding detunings, and are the creation and annihilation operators for the cavity field, and is the atom-field coupling rate, which is determined by the dipole matrix element of the atomic transition, the field per photon within the effective cavity volume , and the cavity mode function at atomic position RDDLSZ (); BHWGS ().
Given an initial configuration , with both atoms in state and zero cavity photons , and large detunings , where is the cavity mode linewidth, we can use second order perturbation theory to eliminate the non-resonant states with a single photon in the cavity. We then obtain that each atom in state experiences a cavity-induced level shift (ac Stark shift) and small level broadening . In addition, state couples to states with rate via virtual photon exchange between the atoms in the cavity BHWGS (); RDDLSZ (); BurImam (); MCGKJS-SPS (). The corresponding interaction Hamiltonian reads
Note that states and are also coupled to state with two photons in the cavity. But due to the large detunings , these transitions yield only small (fourth order) level shifts accounted for below. In second order in , the energy offsets of states , relative to , are , where and . If, by an appropriate choice of (with ), the transitions are made resonant, the Hamiltonian (2) would describe an effective resonant DDI, or Föster process, between a pair of Rydberg atoms and . Then the eigenstates of (2) form a triplet of states and with the corresponding energies and relative to that of state . Note that unlike the free space DDI of LFCDJCZ (), here the DDI has very long range as it is mediated by the cavity field extending over cm.
We next consider the non-resonant case of . Starting from Hamiltonian (1), we use fourth order perturbation theory to eliminate states , and , connected to the initial state via non-resonant single- and two-photon transitions. This yields
which describes an effective cavity mediated VdWI between a pair of Rydberg atoms and with strength . The effect of this interaction is to shift the energy of two atoms simultaneously excited to state by the amount .
Before proceeding, let us estimate the relevant parameters achievable in a realistic experiment. For atoms placed in the vicinity of CPW field antinodes, the coupling constants are approximately the same, (see below). Setting () and , the DDI coefficient is . On the other hand, with and , the VdWI strength is . The total relaxation rate of a Rydberg state () is , where is a small intrinsic decay and is the cavity-induced relaxation. To achieve coherent interactions, we require that , which leads to the condition , tantamount to the strong coupling regime of cavity QED RBHrev ().
For a CPW cavity with strip-line length cm and electrode distance m [Fig. 1(a)], the effective cavity volume is m. The mode functions are 1D standing waves or with being, respectively, an even or odd integer and BHWGS (); MCGKJS-SPS (). Choosing e.g. , the mode wavelength is mm and there are field antinodes. With effective dielectric constant , the mode frequency GHz. For properly selected Rydberg states , the transition frequencies can be adjusted with high precision using static electric and magnetic fields RydAtoms (). The dipole matrix element between neighboring Rydberg states with principal quantum number scales as , which for yields MHz. In a cavity with quality factor , the photon decay rate is KHz, while KHz. Thus the strong-coupling regime with the above stringent condition can be realised for .
Employing a master equation approach QCcomp () with the exact Hamiltonian (1), we have performed numerical simulations of the dissipative dynamics of the full system with above parameters and . Results for the cases realizing the effective DDI and VdWI are shown in Fig. 2 and compared to simulations with the corresponding effective Hamiltonians (2) and (3). As expected, the agreement between the exact and effective models is good for and excellent for . In the case of DDI, for both values of , the decoherence and population losses are very small on the time scale of several oscillation periods between states and . In the case of VdWI, the population loss is appreciable on the much longer time scale of : At time , when state acquires phase shift , its population for , and for . Thus, with realistic experimental parameters and (MHz, KHz and KHz), conditional phase shift of for a pair of Rydberg atoms can be achieved with fidelity . With CPW cavity improvements and parameter optimization, the above fidelity may be further increased.
We envision several quantum information protocols utilizing Hamiltonians (2) and (3) in ensembles of alkali atoms in the ground state. Trapping cold atoms at a distance of -m from the surface of a superconducting chip, incorporating the CPW cavity [Fig. 1(a)], is possible with presently available techniques atchip (); mtrap (); otrap (). A cigar shaped volume would contain atoms at density cm. Each atomic ensemble should be positioned near the cavity field antinode, so that the mode function is approximately constant throughout the atomic cloud. The corresponding coupling constants can then be assumed the same for all the atoms in the CPW cavity.
Employing light storage techniques based on EIT lukpet (); EITrev (), the atomic ensembles in the setup of Fig. 1(a) can serve as reversible quantum memories for single photon qubits. Briefly, in a typical EIT setup, atoms in the ground state resonantly interact with a weak (quantum) field on the transition , while a coherent driving field with Rabi frequency (and wavevector ) couples the excited state to the long-lived (metastable) state . When the light pulse (with wavevector ) enters the EIT medium, it is transformed into the so-called dark-state polariton fllk (); EITrev (), which propagates in the medium with reduced group velocity , where with . The slowing down of the pulse upon entering the medium leads to its spatial compression by a factor of (). Once the pulse has been fully accommodated in the medium, by turning off (), the photonic excitation is adiabatically mapped onto the collective long-lived atomic excitation represented by state which involves a single Raman (spin) excitation, i.e., atom in the metastable state . At a later time, the photon can be retrieved on demand by turning on. Importantly, in order to accommodate the pulse in the medium with negligible losses, the optical depth of the atomic cloud should be large fllk (); EITrev (). With a typical resonant absorption cross-section for the alkali atoms cm, and the above cited density and medium length mm, we have large optical depth .
Using the cavity-mediated DDI (2), we can implement the dipole blockade LFCDJCZ () of multiple Rydberg excitations in an atomic ensemble. This, in turn, can be used to prepare the collective state and subsequently generate single photon pulses, as summarized below. Consider the level scheme of Fig. 3(a), where coherent laser fields with Rabi frequencies and and wavevectors and resonantly couple the lower atomic states and to the Rydberg state . Initially all atoms are in state , and only is on. Then the laser field induces the transition from the ground state to the collective state representing a symmetric single Rydberg excitation of the atomic ensemble. The collective Rabi frequency for transition is . Once an atom is transferred to state , the excitation of a second atom is suppressed by the resonant DDI between the atoms, provided that . Indeed, out of the three eigenstates of (2), the unshifted eigenstate is not coupled to state by , while transitions are shifted away from resonance by and therefore are inhibited. Hence, a laser pulse of area (an effective pulse) prepares the state . The probability of error due to populating the doubly-excited states is found by adding the probabilities of all possible double-excitations, . Additionally, Rydberg state relaxation during the pulse time causes an error . The total error probability is minimized by choosing . By subsequent application of the second stronger laser with pulse area ( pulse), state is quickly converted into , which is precisely the state we need for generating a single photon, as described above and illustrated in Fig. 3(a). For the present parameters and choosing the optimal Hz, the preparation time of state is s with the fidelity .
Note that Rydberg atoms interact also via direct DDI LFCDJCZ (), which, for closely spaced atoms and , is much larger than the cavity-mediated DDI . However, is a short-range interaction, and already at interatomic distances m it is smaller than , whose range is given by the CPW cavity size.
Since the cavity-mediated DDI is present between any pair of Rydberg atoms in the cavity, the above technique can be extended to create an entangled state of any two ensembles and within the cavity of Fig. 1(a). Thus by applying simultaneously to both ensembles, due to the dipole blockade, only one atom will be excited to state . The duration of the pulse should be chosen according to , since it now drives atoms. Using the second laser pulse with area to quickly bring the population of state to the metastable state , an entangled state of atomic ensembles and sharing a single collective spin excitation, will be produced.
We now describe possible uses of the cavity-mediated VdWI (3). As detailed above, atomic ensembles can serve as reversible quantum memories for photonic qubits. Conversely, individual ensembles can represent qubits storing any state of the form in the corresponding superposition of collective states. Consider two such ensemble qubits and in the cavity of Fig. 1(a). A resonant -pulse applied to the transition in both atomic ensembles, , will then convert state of each ensemble to the state with single Rydberg excitation [see Fig. 3(b)]. Since any two atoms in state interact via VdWI with strength , during time they will accumulate a phase shift . Only if both ensembles were initially in state , the above phase shift occurs, since otherwise there will be only zero or one atom in state and the VdWI (3) will not take place. A second -pulse can then convert state of each ensemble back to the original state . Thus the universal cphase gate () QCcomp () would be implemented between and .
The VdWI between Rydberg atoms can also be employed to implement the cphase gate directly between two photonic qubits. A possible setup is shown of Fig. 3(c), where single photon fields and propagate in the atomic ensembles and under the conditions of EIT in the ladder configuration: the quantum fields act on transition , while transition is driven by resonant classical field with Rabi frequency . Upon entering the medium, each quantum field is converted into the dark-state polariton whereby part of the photonic excitation is temporally transferred to the atomic excitation. For , the polariton propagating with group velocity is mainly an atomic excitation. Thus, each field containing a single photon creates in the corresponding ensemble an atom in state . These atoms interact via the cavity mediated VdWI (3) resulting in the cross-phase modulation between the two quantum fields. If both fields enter the corresponding ensembles simultaneously, the interaction time is . Calculations similar to those in DDEIT () show that the non-linear phase shift for a pair of single photons is given by . We can express the group velocity as , where is the coherence relaxation rate on the transition , with typical value for alkali atoms MHz. With the other parameters given above and choosing MHz (), we obtain . Thus the two-photon input state acquires a conditional phase shift of , and more generally, the cphase gate () between two photonic qubits is realized.
To summarize, ensembles of cold atoms trapped in the vicinity of a microwave CPW cavity can strongly interact with each other via cavity mediated virtual photon exchange between optically exited atomic Rydberg states. This system can serve as efficient and scalable platform to realize various quantum information processing protocols with ensemble qubits and single photons.
Acknowledgements.This work was supported by the EC Marie-Curie Research Training Network EMALI.
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