Rotation sensing with trapped ions

We present a protocol for using trapped ions to measure rotations via matter-wave Sagnac interferometry. The trap allows the interferometer to enclose a large area in a compact apparatus through repeated round-trips in a Sagnac geometry. We show how a uniform magnetic field can be used to close the interferometer over a large dynamic range in rotation speed and measurement bandwidth without losing contrast. Since this technique does not require the ions to be confined in the Lamb-Dicke regime, thermal states with many phonons should be sufficient for operation.


Introduction
The Sagnac effect can be used to measure the rotational velocity Ω of a reference frame by observing the phase shift of an interferometer in that frame whose paths enclose an area A perpendicular to any component of Ω (see, e.g. [1] for a review). The rotationinduced phase shift is given by where A is the vector area enclosed by the two paths. E is the total energy of the particles that are interfering, defined using the relativistic energy-momentum relation For photons, E =hω optical , whereas for atoms of rest mass m moving at non-relativistic speeds (p ≪ mc), E = mc 2 . The sensitivity of a gyroscope is the minimum detectable rotation rate within a detection bandwidth ∆f . For a shot-noise-limited interferometer that detects the outcome of individual interference events at a rateṄ , the uncertainty in the measured phase after running for a time t = 1/∆f will be δφ ≈ ∆f /Ṅ. The sensitivity is given by where the scale factor is given by and we have assumed an orientation such that A·Ω= AΩ for algebraic simplicity. Two primary methods are frequently employed to boost the sensitivity of interferometric gyroscopes. For photons, optical fibers (or ring laser cavities) allow many effective round-trips through the Sagnac interferometer, thereby increasing the effective area A by 2 times the number of round trips (M) without increasing the actual area of the apparatus. This, coupled with the largeṄ possible in these devices, leads to a state-of-the-art reported sensitivity of S = 1.2 × 10 −11 rad/s/ √ Hz, achieved by a ring laser with 16 m 2 enclosed area [2].
Another approach is to use atoms on ballistic trajectories instead of photons, which increases E by a factor of mc 2 /hω optical ≈ 10 11 . The drawbacks of this approach, as compared to an optical gyroscope, are thatṄ is smaller, and the free-flight atom trajectories enclose the interferometer area A only once. The latter constraint has meant that increasing A has necessarily involved increasing the physical size of the apparatus, which can be undesirable for some applications. Furthermore, long atom trajectories and large separations make the measurement susceptible to systematics that can produce path-dependent phase shifts, such as magnetic field gradients. Nonetheless, the improvement in E has enabled atom interferometers to demonstrate high rotation rate sensitivity, with demonstrated state-of-the-art short-term sensitivities of S = Figure 1. Trajectories of an ion during interferometer operation in (a) position space, (b) momentum space (c) x phase space and (d) y phase space. The ion's starting coordinates are indicated by a circle, and the trap center after the y-displacement in step (iii) is indicated by an ×. Red and blue curves represent the trajectory for the two spin states. Trajectories for different starting conditions are qualitatively similar to these, with the exception that for a ground-state ion, the trajectories for the two spin states completely overlap.
Here, we show that trapped atomic ions provide a way to use both methods simultaneously to increase the interferometer sensitivity. While interferometers with enclosed area have been demonstrated with clouds of trapped neutral atoms [5,6], maintaining the coherence across the ensemble needed for a gyroscope has proved difficult. Here we introduce a combination of laser-driven spin-dependent momentum kicks in one direction with ion trap voltage changes along an orthogonal direction that perform interferometry with trapped ions in a Sagnac (as opposed to Mach-Zehnder) configuration. This allows atomic trajectories to repeatedly enclose the same area, thereby accumulating Sagnac phase continuously for a time that is not limited by a ballistic flight trajectory. Since the enclosed area is proportional to the displacement along both directions and only one of these needs to be state-dependent, the interferometer area can be increased with trap voltage alone, circumventing the need to drive more coherent momentum transfer from the laser. The harmonic trapping potential makes the area enclosed independent of the initial ion velocity, eliminating a source of scale factor instability found in free space atom interferometers. These factors, coupled with the extremely long coherence times of trapped ions, gives the trapped ion interferometer the potential to enclose a large effective area in a small apparatus with high stability.

Interferometer operation
We begin by introducing the protocol for measuring rotations with a single ion that hosts a qubit with internal states |↑ and |↓ . As shown in Fig. 1, the enclosed area will be in the x, y plane, and the (secular) trap frequencies for the ion in these two directions after the y displacement (see below) will be assumed to be degenerate: ω x = ω y ≡ ω. (x, y, z) will be coordinates in real space, while (X, Y, Z) will denote axes of the qubit's associated Bloch sphere. We will assume that the confinement in the z-direction is strong (ω z ≫ ω) so that the system can be approximated as being 2D. The time-sequence of the trapped ion gyroscope proceeds in the following steps (see also Fig. 1 and 2): (i) Prepare the ion in |↓ and apply a π/2 pulse of microwaves about −Ŷ.
(ii) Apply N k spin-dependent kicks (SDKs) in the x-direction (∆p = −N kh ∆kσ Z ) to separate the atom in momentum space.
(iii) Apply a step function in electrode voltages to non-adiabatically displace the trap only in the y-direction a distance y d .
(iv) Allow the ion to oscillate in the trap for an integer number (M) of round trips ∆t = M2π/ω.
(v) Reverse step (iii) by non-adiabatically switching the trap voltages back to their original values.
(vi) Reverse step (ii) by applying N k SDKs in the other direction (∆p = N kh ∆kσ Z ) to close the interferometer.
(vii) Apply another π/2 pulse with microwaves about an axis inclined by φ in the X, Y plane from the −Ŷ axis of the Bloch sphere, then measure the internal state of the ion in the qubit basis.
We note that steps (i) and (vii) are a standard Ramsey sequence, so qubit and microwave oscillator coherence are required for the duration of the protocol. Even for non-clock-state qubits with magnetic sensitivity on the order of a Bohr magneton (µ B ), qubit coherence times of the order 1 s or greater can be achieved [7]. We will describe the details of the gyroscope protocol assuming the magnetic field on the ion is zero before discussing the magnetic field effects.

Phase-space displacements
The trapped ion gyroscope relies on two different methods to produce displacements in motional phase space: spin-dependent kicks that transfer photon momenta to the ions in directions that depend upon the ion's spin (qubit) state, and trap voltage steps that rapidly displace the trap center. Since a displacement operation in phase space necessarily involves a large number of Fock states, both of these operations take place much faster than the resolved-sideband limit (T ≈ 2π/ω), and can be thought of as driving many different motional transitions at once.
The spin-dependent kicks [8] of steps (ii) and (vi) act as the beam splitters in the matter-wave interferometer. The speed of the SDK is enabled through the utilization of "ultrafast" mode-locked lasers to transferh∆k of momentum to the ion (see [8,9] for more detail). Conceptually, the ideal SDK transfers a momentum kick to the ion whose direction is reversed for the two spin states via the operator whereD x [s] displaces a coherent state in x phase space a distance s (see Fig. 1 and 2) and η is the Lamb-Dicke factor for the laser-ion interaction in the x-direction (η ≡ ∆kx 0 = ∆k h/2mω). Since step (vi) drives the same process as (ii) with the direction of the kicks reversed (effectively replacing i → −i in (5)), we have suppressed the laser beat note phase when writing (5) since it plays no role as along as it is stable during a single enactment the interferometer protocol. The qubit raising and lowering operators (σ ± ) flip the spin state of the qubit, so one way that larger displacements (i.e. M of them) can be made is by repeating this operation after a delay by half a motional period [10]. For algebraic simplicity, we will assume for our protocol that the number of spin-dependent kicks applied (N k ) is even and that an extra half-period of motion is inserted after the last kick to preserve the harmonic oscillation phase of the initial motional state. Working in the coherent state basis for describing the ion's motion in x and y (denoted by coherent state parameters α x and α y ), step (i) results in the state The SDKs in step (ii) induce spin-orbit coupling to produce the entangled state where IR(α) denotes the real part of the coherent state parameter α.
Interferometers based on SDKs have been proposed [11] to measure the Sagnac effect, and have recently been implemented in a 1D non-Sagnac geometry to measure temperature over a wide dynamic range [10]. However, for a Sagnac gyroscope, the second displacement need not be spin-dependent and can therefore be implemented as a simple trap center shift. Electrode voltages can be rapidly changed to displace the trap center, an operation that has been demonstrated to couple to thousands of Fock states to perform a coherent state displacement operation [12]. Compared to coherent momentum transfer from a laser, voltage-driven motion of this sort can produce a larger displacement, and does so without the detrimental effects of spontaneous emission and differential AC Stark shifts associated with laser-driven gates.
A rapid shift of the trap center by a physical distance y d along y can be modeled with a displacement operator where y 0 = x 0 ≡ h/2mω and we will refrain from writing phase terms that for this protocol are global.

Rotation-induced phase
The effect of rotation in this system can be described in either the non-rotating frame (the ion's frame) or the rotating reference frame (the apparatus frame). We choose the former, which means that the rotation manifests itself as change in the direction of the kicks in steps (v) and (vi). A constant rotation rate Ω about the positive z-axis of the apparatus will shift the angles of these kicks by This transforms the displacement operators according tô and the state of the ion after step (vi) is where the relative phase (δ) is given by (11) shows that this protocol leaves residual entanglement between the spin and motion in both x and y. Using |µ i,f (θ) to denote the final motional states in x and y for the parts of the wavefunction that are associated with spin state i ∈ {↑, ↓} in (11), the overlap is where the first term comes from the imperfect state overlap (which is confined entirely to momentum space) and the second is a pure phase term called the overlap phase δ ′ : Residual entanglement between spin and motion will reduce the contrast of the interferometer, and it is the sum of δ and δ ′ that contributes the phase shift that is measured using this protocol. However, as we show below, the phase shift that is measured is unaffected by the initial ion temperature, and there is no requirement that this device be operated in the Lamb-Dicke regime.

Readout
In order to measure the rotation-induced phase (δ + δ ′ ), step (vii) applies a second Ramsey zone with a controllable phase shift φ, yielding This step maps the motional phase onto the internal state of the ion, which would then measured using standard fluorescence techniques. The probability of measuring, for instance, spin up (|↑ ) is given by where and P (α j ) is the Glauber-Sudarshan P -representation describing the (potentially mixed) initial motional state in the coherent state basis. A(α y ) is the classical, geometric area of the ellipse enclosed by the ion trajectories in the x, y plane, which depends upon the initial position in y via y i = 2y 0 IR(α y ): In Appendix A, we show that a semiclassical derivation agrees with this quantum calculation. If we expand the argument of the cosine in (17) to first order in θ, we see that it simplifies to φ − Φ, where This is identifiable as the Sagnac phase shift (1) with an effective area of A eff = 2MA(α y ) since the ion encloses the ellipse area A(α y ) twice each period for M periods. (19) also provides some insight into the origin of the scale factor of this interferometer: the rotation angle θ is effectively "amplified" by a gain factor of A(α y )/πx 2 0 , the ratio of the enclosed area to the area of the ground-state wavefunction. This gain factor is the angular momentum of the ion's motion divided byh, and the interferometer can therefore be thought of as a generalized atomic or nuclear spin gyroscope with a very large effective spin.

Finite temperature
For an ion that is pre-cooled to the motional ground state along y (α y = 0), (19) gives precisely the desired outcome (1) for the trapped ion gyroscope. For an ion that is initially in a thermal state with mean phonon occupation numbersn x =n y ≡n, (16) can be used to calculate the probability of measuring spin up: which is valid to all orders in θ. The effect of finite temperature is a reduction in the contrast of the interference, but does not produce a phase shift of the signal. However, since the exponent in (20) is proportional to sin 2 (θ/2) and the Sagnac phase shift is proportional to sin(θ), the free evolution time (∆t in (9)) can be chosen to satisfy and the interferometer can be operated at essentially full contrast, even at high temperature. There is therefore no requirement that the ion be cooled to the Lamb-Dicke regime, and as we estimate below, Doppler cooling should be sufficient for full-contrast operation.

Magnetic field effects
Since the ion is moving while it is accumulating rotation-induced phase, a nonzero magnetic field will give rise to a Lorentz force on the moving monopole. For a magnetic field in the z-direction, this will cause the ion's orbit to precess in the x, y plane, which will lead to a false rotation signal. Specifically, the magnetically-induced rotation rate (Ω m ) can be found [13] by equating the Lorentz and Coriolis forces for a uniform, static magnetic field in the z-direction (B = B zẑ ): This rotation rate is half the cyclotron frequency ω cyc = eB/m = 2Ω m , and the precession angle this will produce is boosted up by the gain factor of A(0)/πx 2 0 to give a magnetically-induced phase shift of This phase shift can be interpreted as the dynamical phase from the Zeeman shift of the ion's motional magnetic moment, where I ≡ eω/2π is the current from the ion's motion. This matches the classical expression for the magnetic moment of a current loop of area A(0).

Non-harmonic corrections
The analysis we have presented has thus far assumed a perfectly harmonic potential. We can find the first-order phase correction for small non-harmonic terms of the potential by treating these terms as a perturbation and integrating over the unperturbed trajectories. Assuming the potential remains separable the formulas below hold for each axis. Let us write the general potential as where x l is a length scale for the amplitude of the ion's motion and the {C i } are dimensionless numbers assumed to be much smaller than one. With a harmonic trajectory x(t) = x l sin(ωt + φ), only even i terms are non-zero. Integrating over M orbits gives

Performance
Once the evolution time has been fixed, the sensitivity of the trapped ion gyroscope can be written Since this is independent of the trap frequency ω (we assume M ≫ 1 and can therefore be chosen essentially arbitrarily), the trapped ion gyroscope can be operated in a relatively low-frequency trap as compared to typical traps for applications requiring resolved sideband operations. This provides the practical advantage of making the nonadiabatic operations easier to achieve with high fidelity in a fixed time. It also permits the use of a trap whose electrodes are far apart and far from the ion, which will suppress surface-induced heating and patch charge perturbations and improve harmonicity for a fixed absolute length scale.
We also note that the performance of this rotation sensor is independent of the mass of the ion, and depends essentially only on the wavelength of the laser used to drive the SDKs. We will estimate parameters for 171 Yb + , which was used for the first demonstrations of spin-dependent kicks [8], but estimates for other species will be similar in magnitude.
Since this qubit has a demonstrated coherence time exceeding 1000 s [16], a freeevolution time of ∆t = 1 s should be straightforward to achieve. For a (secular) trap frequency of ω/2π = 10 kHz, a trap displacement of y d = 100 µm would correspond to n ≈ 9 × 10 5 phonons, where displacements corresponding to n ≈ 10 4 have already been demonstrated [12]. Cooling Yb + to the Doppler limit (T D =hγ/2k B ) in such a trap will result in an interference contrast of 85% for the (sidereal) rotation rate of the earth Ω e ≈ 73 µrad/s. A trapped ion gyroscope operated with these parameters would have a scale factor of ∂Φ/∂Ω = 52 rad/Ω e and a sensitivity of S = 1.4 × 10 −6 rad/s/ √ Hz. Improvements in the numbers of SDKs or the distance of coherent trap displacements would make this competitive with cold atom interferometers that use large numbers of atoms.
For an interferometer using the parameters discussed above, the magneticallyinduced rotation rate per unit field is Ω m /(2πB z ) = 4.5 Hz/G. The associated motional magnetic moment is µ m = 1.1µ B , where µ B is the Bohr magneton. Since the magnetic field stabilization required to combat this systematic only needs to be applied to a small volume (≪ 1 cm 3 ), this magnetic sensitivity resembles the effect of using a Zeemansensitive qubit, and many of the technical difficulties associated with this have been overcome in various trapped ion quantum information processing experiments [7].
In the limit where the magnetically-induced rotation rate is much slower than the (secular) trap frequency (Ω m ≪ ω), the ion's motion is in elliptical orbits of fixed area whose orientation slowly rotates. Too much rotation will reduce the contrast of the interference signal since the kicks in step (v) and (vi) will not efficiently close the interferometer loop. However, trapped ions have also been demonstrated as superb magnetic field sensors, and it seems likely that with a periodic measurement of a stationary ion's Zeeman splitting, a well-controlled field could be applied to cancel this effect.
Likewise, magnetic rotation could be leveraged to cancel the contrast reduction associated with high actual rotation rates (the exponential factor in (17)). In this "closed-loop mode," the magnetic field needed to cancel the rotation would become the output signal for the interferometer, and low-resolution rotation sensors could be incorporated to feed forward the magnetic field needed to keep the interferometer contrast maximized and on the steepest part of a fringe.

Discussion
As compared to free-flight matter-wave interferometers, the trapped ion device provides many practical advantages. First, the physical size of the interferometer can be compact while still retaining a large effective interferometer area by using multiple orbits. Second, since the ion wavepacket re-combines in space twice per trap period, this interferometer can be interrogated over a wide dynamic range of free-evolution times. Fast rotation rates, which can be problematic in neutral atom systems if the wavepackets don't recombine or leave the interferometry region, can be compensated by applying uniform magnetic fields. The operational mode could be to actively stabilize the fringes with an applied field, which becomes the readout signal. There is also no need to keep multiple optical beam paths interferometrically (relatively) stable since the only steps that are sensitive to a laser phase (the SDKs, steps (ii) and (vi)) are driven by the same laser with its beam traversing the same optical path. In addition, by using single ion wavepackets which travel the same average trajectory, only in opposite directions, we eliminate spatially varying systematics. Finally, free-flight interferometers have sensitivities to accelerations and the atomic beam velocities, whereas the scale factor for the ion trap interferometer depends only on the momentum kicks and trap displacement.
Another advantage of using trapped ions instead of neutral atoms for matterwave interferometry is the potential to leverage the advances in trapped ion quantum information processing to produce sub-shot-noise scaling of the sensitivity with ion number. For example, a collection of N I ions could be prepared in step (i) in a GHZ spin state, |ψ = 1 √ N I (|↓↓↓ · · · ↓ + |↑↑↑ · · · ↑ ) , and the same protocol could be used as for the single ion to accumulate phase, but with the resolution (and sensitivity) enhanced by a factor of N I . These states (29) have been created for as many as N I = 14 ions [17], and multiple groups are actively pursuing various ways to scale up the size of entangled trapped ion systems.