Entangling the lattice clock: Towards Heisenberg-limited timekeeping
We present a scheme for entangling the atoms of an optical lattice to reduce the quantum projection noise of a clock measurement. The divalent clock atoms are held in a lattice at a “magic” wavelength that does not perturb the clock frequency – to maintain clock accuracy – while an open-shell “head” atom is coherently transported between lattice sites via the lattice polarization. This polarization-dependent “Archimedes’ screw” transport at magic wavelength takes advantage of the vanishing vector polarizability of the scalar, , clock states of bosonic isotopes of divalent atoms. The on-site interactions between the clock atoms and the head atom are used to engineer entanglement and for clock readout.
pacs:37.10.Jk, 03.67.Bg, 06.30.Ft
Quantum entanglement is a crucial resource in quantum computing and has the potential to improve precision measurements NielsenChuang00 ; Preskill00PrecisionMeasurement . Here we propose a scheme for entangling an ensemble of several thousands of neutral atoms, with the specific goal of demonstrating the power of entanglement for measuring time.
Measuring time with atoms relies on the fact that the quantum-mechanical probability of making a transition between two clock levels depends on the detuning of the probe field from the atomic transition frequency . By measuring the probability as a function of , one can infer if the two frequencies are equal and thereby “lock” a local oscillator to the atomic transition. Counting the number of oscillations of the local oscillator tells time. The precision of measuring is limited by the quantum projection noise PhysRevA.47.3554 . For a measurement of unentangled atoms the resulting signal-to-noise of scales as : the standard quantum limit (SQL). The use of entanglement holds the promise of improving clock precision to the Heisenberg limit, with signal-to-noise scaling as .
Measurements with uncertainty below the SQL may be achieved with squeezed atomic states Mandel98QNDSqueezing ; VuleticQNDclock ; YeQNDproposal . While this technique can address large number samples of atoms, squeezing experiments have attained signal-to-noise ratios far from the Heisenberg limit. In other work, clock measurements at the Heisenberg limit have been demonstrated for small numbers of entangled ions in traps Wineland04 . Those experiments created maximally entangled states to achieve measurements at the Heisenberg limit via a Ramsey-type measurement protocol: a generalized pulse creates a Greenberger-Horne-Zeilinger (GHZ) state, the atoms undergo free evolution, and a final generalized pulse is used for readout. However, scaling ion traps up to entangle increasingly larger number of ions currently remains a work in progress.
Using atom-atom interactions to engineer entanglement between neutral atoms trapped in a lattice may offer the best of both worlds: maximally entangled large-number samples. Previous proposals have noted the virtues of using alkaline-earth-like atoms in lattices for quantum information and quantum computing DerCan04 ; HayJulDeu07 ; GorReyDal09 ; DalBoyYe08 ; ShiKatYam09 ; here we focus on entangling atoms for improving the atomic clock.
In optical lattice clocks, millions of divalent atoms (such as Sr or Yb) are trapped in an optical lattice (register) operating at a “magic” wavelength . At this wavelength, both clock levels, and , are shifted by the lattice lasers equally, so that the clock frequency remains unperturbed KatoriMagic03 . These clocks have demonstrated long coherence times and have already realized great improvements in both accuracy and precision over the current primary frequency standard A.D.Ludlow03282008 ; T.Rosenband03282008 . Notably the signal-to-noise ratio of current-generation clocks is approaching the SQL A.D.Ludlow03282008 , so an “entangled clockwork” may be of practical benefit.
We would like to entangle a string of strontium clock atoms. Each Sr atom occupies an individual lattice site as shown in Fig. 1. To maintain the clock accuracy we require all lattice fields be at the magic wavelength of strontium. Entangling the atoms using short-range atom-atom interactions requires the transport of atoms between lattice sites. While coherent transport in optical lattices has been demonstrated before using lattice polarization PhysRevLett.82.1975 ; porto07 ; PhysRevLett.91.010407 ; Weiss04transport , such techniques would not work for the clock states of bosonic isotopes, which have a scalar () nature: the optical potential does not depend on the polarization (be it circular or linear) of the lattice lasers. Instead, we use a single “head” atom, which is transported from site to site using the optical polarization PhysRevA.70.012306 .
Lattice — Consider the superposition of two spatially-displaced standing waves of opposite circular polarization . The resulting E-field reads
where winding the phase similarly to the “Archimedes’ screw”controls a relative displacement between the nodes of the two standing waves. The resulting optical potential reads
For an atom in an state (with the quantization axis taken to align with the lattice lasers)
Here and are frequency-dependent scalar and vector (axial) polarizabilities. Neglected tensor contribution is suppressed BelDerDzu08Clock for atoms.
The two clock states and will experience the same trapping potential; at the two ac polarizabilities are the same. Note that the vector part of the polarizability is zero for the scalar clock states. If two states of the head atom and have different vector polarizabilities, they will see different potentials. For an appropriate choice of lattice parameters, discussed below, the state couples preferentially to the permanent sub-lattice, while couples preferentially to the moving sub-lattice. Unfortunately, the commonly employed lin lin transport lattice algorithms PhysRevLett.82.1975 ; porto07 ; PhysRevLett.91.010407 ; Weiss04transport cannot be employed here directly, as the potential will wash out for the clock atom in the lin lin configuration. However, for appropriate choice of lattice intensities (discussed below), the clock atoms remain pinned to the permanent sub-lattice. This state-selective transport enables the entanglement of the clock atoms with the head atom state, as shown in Fig. 1 and described below.
Clock protocol — We describe the system as the product of the state of the clock register and the state of the head atom. For example, a possible basis state of clock atoms and a head atom in the spin-up state is We require two gates: a single-qubit Hadamard gate (an analog of a pulse) and a two-qubit phase gate. No individual addressing is required. The phase gate involves the “head” atom state-selectively transported to overlap with the target clock atom at position in the lattice register, and .
A practical realization of the Hadamard gate involves interaction with a near-resonant pulse of optical frequency for the clock qubits and a near-resonant pulse of microwave frequency for the head atom. We want to measure the clock frequency by tuning the driving frequency . Below we show that, as in the conventional Ramsey-type clock frequency measurement, the probability of making a clock transition depends on the detuning , allowing to zero-in on the clock frequency. Theoretical analysis is simplified by transforming into a rotating reference frame; the relevant chain of operators for the compound wavefunction involves a product of clock-state rotation operators at frequency and a rotation operator for the head atom at frequency .
We start by filling a 1-D lattice with a single head atom and clock atoms, Next we apply the Hadamard gates to the head atom and to all the clock atoms:
where is the computational (binary) basis set for the clock register. In the next step we use the transport lattice and move the state of the head atom along the clock register to perform a collisional phase gate at each site.
We presume that the and clock states will have different scattering lengths for their interaction with the state of the head atom, and thus a different mean-field interaction. The transport lattice will be moved in such a way that the head atom will remain on-site with each clock atom for a sufficient period of time to produce a relative phase shift of PhysRevLett.82.1975 . This generates the entangled wavefunction
Here , i.e., it is the number of raised bits in the binary representation of . By applying the Hadamard gate to the clock register we obtain the GHZ state
The entire procedure may be considered as a generalized pulse in the space of maximally-polarized states.
As in traditional Ramsey spectroscopy we let the GHZ state evolve freely for a time . In the rotating reference frame, a phase is accumulated during the free evolution. Notice that is the detuning for the optical clock frequency, while is the detuning for the microwave transition of the head atom. Therefore Finally, we repeat the generalized pulse, arriving at
where the -enhanced phase is encoded into the state of the microwave head qubit.
For readout of the clock, the state of the head qubit can be read directly via laser-induced fluorescence. In the event that the light collection efficiency is insufficient to read out the state with high efficiency IonClockReadout , the state of the head atom can be transferred to many alkaline earth atoms (as done in the first stage of the entanglement algorithm) for efficient readout.
Choice of the “head” atom and lattice parameters — We start with a detailed analysis of the transport lattice which dictates the choice of the head atom. We introduce the fractional intensity misbalance for the two circularly polarized sublattices of Eq. (1): . It does not depend on atomic properties and . For the clock atoms, the minimum depth of the optical potential — which occurs at , or positions (b) and (d) in Fig. 1 — is proportional to . However, the larger is, the weaker the lattice becomes for one of the head atom states. The fixed value of will be used in all the following calculations.
We further introduce a ratio of the vector and scalar polarizabilities,
The ability to translate the state while holding the state stationary is determined by this ratio. For example, for , and the atom does not see the stationary lattice. Similarly, for the atom has no coupling to the moving lattice. In general, for a positive value of we must satisfy the criteria and together with .
Qualitatively, for the head atom transport one needs . It turns out that at the magic wavelengths specific to divalent atoms (see Ref. DerObrDzu09 ), none of the commonly used alkali-metal atoms satisfies this constraint. Alkali atoms have the ground states and the vector polarizability (rank 1 tensor) arises only due to relativistic effects. Fortunately, atoms with ground states (group III) have large vector polarizabilities. Aluminum is a suitable choice for the head atom.
Al has a ground state. The nuclear spin of 5/2 gives rise to two hyperfine structure levels: and , separated by 1.5 GHz. Cooling Al has already been demonstrated McGGilLee95 with the goal of atomic nanofabrication. The laser cooling was carried out on the closed transition with the recoil limit of 7.5 K. Once trapped, the atoms can be readily transferred from the metastable cooling state to the ground (head) state. Lattice-trapped Al was also considered for quantum information processing RavDerBer06 and for a microwave lattice clock (microMagic clock) BelDerDzu08Clock .
To evaluate the dynamic polarizabilities for Al, we employed ab initio methods of relativistic many-body theory. To improve on the positions of atomic resonances, for low-lying energy levels we replaced the ab initio energies with experimental values. The resulting dynamic polarizabilities of Al are shown in Fig. 2.
We focus on the promising case of the Sr “blue” magic wavelength TakKatMar09 at nm. Here the clock atoms are confined to minima of laser intensity, reducing photon scattering. Scalar polarizabilities are and Al offers many viable states to implement transport: or (, respectively) and or (), for example. We choose and .
Due to being “blue-detuned” for both Al and Sr, the atoms will be confined to regions of intensity minima. While the transport lattice provides axial confinement, the radial confinement is provided by two transverse “blue-detuned” magic lattices with polarization parallel to the transport lattice’s k-vector (to avoid interference with the transport lattice). The transverse blue lattices create a series of tubes Bloch05LatticeReview , each containing a transport lattice. This provides tight radial confinement and enables many transport lattices to be run in parallel. We note that to prepare the initial states in the resulting 3-D lattice, both species can be prepared in the ground vibrational states of individual lattice sites using 3-D Raman sideband cooling PhysRevLett.84.439 ; PhysRevLett.85.724 .
The depth of the optical potentials is critical, as it determines the rate of unwelcome diffusion out of sites DerCan04 . The depth varies with the displacement phase . For our choice of , the depth is weakest when it is on-site with Sr. At this maximum-overlap position, and , where and is the speed of light. We require , where is the recoil energy for an atom of mass . This translates into a minimum intensity of , determined by the lighter Al.
Gate times and decoherence — The number of clock atoms which may be entangled will be limited by decoherence and phase-gate times. The main sources of decoherence are anticipated to be inelastic collisions between the clock atoms and the head atom and light scattering.
The decoherence rate due to the photon scattering is . Here is a suppression factor DerCan04 accounting for atomic wave functions being centered at zero intensity. We find for Sr and for Al.
To estimate the time required for a phase gate operation, we note the interaction energy of two particles in overlapped ground states of independent 3-D anisotropic harmonic potentials is given by where is the scattering length, is the reduced mass, and is defined analogously to the reduced mass with being trap frequencies.
The axial trap frequencies may be determined from the parameters above ( etc.) at overlap. The radial trap frequencies are determined assuming the transverse lattices are operated at the same intensity as the transport lattice. Estimating the difference in the excited-state and ground-state scattering lengths to be yields an estimated gate time of and a transport time of s.
Conclusions — With the pessimistic assumption that a single photon scattering event will decohere all the clock atoms, we expect that – within each 1-D lattice – one would be able to put Sr atoms into the maximally entangled GHZ state with high probability. This would enable a reduction in the projection noise of lattice clocks. Moreover, we note that many of the usual requirements for producing highly-entangled states between atoms – such as single-site addressability, single-site readout, and unity site occupation – are absent in this scheme.
Possible improvements would include the use of entangled states less sensitive to photon scattering as a source of decoherence. Moreover, it is possible that combining these techniques with more sophisticated gate operations could lead to the development of a full quantum computer. In that case, error correction techniques could potentially be used to further increase the number of atoms while maintaining high fidelity.
But even without these improvements, we note that this scheme occupies an interesting “middle-ground” of experimental schemes for clock entanglement. It holds promise for use with larger numbers of atoms than has been demonstrated to date with ion traps. And while it cannot entangle as large-number samples as are used in spin-squeezing experiments, it may be able to produce greater levels of entanglement.
Unanswered questions remain, such as the value of the Al–Sr scattering length (which determines the gate time) and the rate coefficient for inelastic Al–Sr collisions (which is an additional source of decoherence). With 3 naturally occurring bosonic isotopes of Sr, it is likely that a favorable scattering length can be found. The inelastic rate is unknown, but we note that measurements of Al-group atoms in the ground state have observed slow hyperfine relaxation in collisions with noble gas atoms Weinstein2009InGa .
Acknowledgements — We thank V. Vuletic, J. Ye, and A. Gorshkov for useful comments. This work was supported in part by the US NSF and by the US NASA under Grant/Cooperative Agreement No. NNX07AT65A issued by the Nevada NASA EPSCoR program.
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