Systematic Effects in 2D Trapped Matter-Wave Interferometers

I shall describe the evolution of the interferometer’s wavepackets in the basis of coherent states. First, I consider the case of a 1D interferometer. The result is easily extended to higher dimensionality. A harmonically confined particle is assumed to be initially in a coherent motional state (extension to thermal states is considered later) and a pure spin state:

I work in an interaction picture such that phases associated with the time-evolution of the internal states are not considered. The coherent state $\ket{\alpha}$ is defined by a complex number $\alpha$ that describes the expectation values of position and momentum:

where $m$ is the mass of the particle and $\omega$ is the trap frequency. The state $\ket{\alpha}$ can be expressed in terms of Fock states (‘number’ states), $\ket{n}$, as

The internal state is prepared in a superposition by application of a $\pi/2$ pulse:

A series of state-dependent kicks (SDKs) are performed at time $t=0$, splitting the wavepacket into two and entangling the spin and motion, generating two distinct momentum states. This can be represented by the following operator:

where $n_{\rm k}$ is the number of kicks applied and $\eta$ is the associated Lamb-Dicke parameter, given by

where $k$ is the effective wavevector for the transition. The displacement operator, $\mathcal{D}$, is defined as

Following the SDKs, the wavepackets evolve freely in the trap for some time $t_{\rm free}$. In the ideal case, the evolution time is a multiple of the trap period, $\tau=2\pi/\omega$, and the wavepackets return to their initial positions in phase space. Imperfections which lead to a residual separation of the wavepackets cause a reduction in visibility. I represent such an effect as an additional spin-dependent phase-space displacement of the wavepackets:

where in general $\delta_{\downarrow}\neq\delta_{\uparrow}$. After the free-oscillation period, an SDK opposite to the first and a second $\pi/2$ pulse are performed. The final state of the system is thus given by

Given a calculation of the final state of the wavepacket, one can calculate the probability of being in a particular spin state (the signal of such an interferometer) as

It is then straightforward to extract the interferometer phase and visibility.

The same procedure can be applied to the case of a 2D interferometer, in which case the initial state is written as

where $\alpha_{x}$ and $\alpha_{y}$ are the coherent states for the two orthogonal spatial axes. After preparing a superposition of internal states, a real-space displacement, $\mathcal{D}^{y}[\tilde{d}]$, is applied such that the subsequent splitting of the wavepackets occurs at a position $\tilde{d}\sqrt{\frac{2\hbar}{m\omega}}\hat{y}=d\hat{y}$. This displacement is orthogonal to the previously described momentum kicks which are assumed to be along $\pm\hat{x}$. Following recombination of the wavepackets, the initial real-space displacement is reversed ($\mathcal{D}^{y}[-\tilde{d}]$). The upper-left plot of Fig. 1 illustrates these displacements and the resulting orbital motion of the wavepackets. The final state of the system can thus be calculated as

Note that the imperfections $\delta_{\uparrow}$ and $\delta_{\downarrow}$ will in general be in both spatial axes. Again, an expression for the internal state distribution can be extracted that allows for calculation of the associated phase and visibility.

The procedure outlined above assumes a particle initially in a pure coherent state. Extension to a (mixed) thermal state is provided by describing the initial state by the Glauber-Sudarshan P representation. The density matrix of the initial state of the system is given by

where $\bar{n}$ is the mean occupation number. The internal state distribution after the interferometry sequence can thus be extracted as

While the foregoing description provides a rigorous quantum mechanical treatment of the interferometer sequence, considerations of some systematic effects, such as e.g. anharmonicity of the confining potential, are more easily incorporated by adopting a semi-classical treatment to compute the wavepacket trajectories and the associated phases. This treatment is valid provided any anharmonicities are sufficiently small so as to not produce significant distortion of the wavepackets. A thorough analytic treatment of this range of validity is outside the scope of this work, but explicit numerical simulations have been performed which strongly support the validity of the approximation for the parameter range discussed here. In particular, calculations indicate that significant wavepacket distortion is observed after one trap period of evolution when the anharmonic energy shift is the same order of magnitude as the harmonic potenital. For a ${\sim}1~\%$ anharmonic shift, the overlap between the exactly calculated wavepacket and the coherent state approximation deviates at the level of ${\sim}10^{-5}$. Given the ${\sim}10^{-4}$ anharmonic contribution considered here, any wavepacket distortion will be entirely negligible. Thus the coherent state formalism described above can be applied to compute the corresponding visibility (Sec. II.2.2) given trajectories computed semi-classically.

The wavepacket trajectories are calculated by considering a particle moving classically under the influence of a general potential of the form

where I have defined $\tilde{x}=x/R$, $\tilde{y}=y/R$ with $R$ the characteristic size of the system under consideration (I take this to be the magnitude of the initial real-space displacement from the trap centre, $d$). In common experimental configurations, such as e.g. a linear Paul trap, terms of the form $xy^{3}$ and $x^{3}y$ are unlikely to arise due to symmetry considerations. For this reason, and for the sake of brevity, I omit these terms from the results presented here, but their effect was found to be very similar to that of the other anharmonic terms. The size of the anharmonic terms considered is discussed later.

The acceleration is calculated as $\vec{a}(t)=-\vec{\nabla}V(x(t),y(t))/m$ and the equations of motion are numerically integrated. In the ideal case, $c_{1}=c_{2}$ are the only non-zero coefficients and the wavepacket trajectories can be expressed analytically. Equations 18–24 describe the trajectories while incorporating the following imperfections (cf. Fig. 2): $\theta_{\rm com}$, a common misalignment of the momentum kicks from the nominal axis defining the kick directions, while keeping the kicks opposite to each other; $\theta_{\rm rel}$, a relative misalignment of the momentum kicks with respect to each other; $\theta_{\rm disp}$, a misalignment of the initial displacement; $v_{\rm com}$, a velocity component common to both wavepackets and $\omega_{x}\neq\omega_{y}$, a mismatch of the trap frequencies.

The $\pm$ superscripts label wavepackets initially moving in $\pm x$. $v_{\rm SDK}$ denotes the magnitude of the velocity imparted by the SDKs that split the wavepackets, i.e.

Note that the expression for the initial velocity $v_{y}^{\pm}(0)$ is not explicitly written, but can be readily calculated given the imperfections present.

At each point in the trajectory, a corresponding coherent state can be calculated as (cf. Eqs. 2 and 3)

The direct analogy between the classical trajectory and the associated coherent state facilitates quick calculation of the effects of experimental imperfections.

With the wavepacket trajectories in hand, one can use the path integral formulation Sakurai and Napolitano (2017); Pippa Storey and Claude Cohen-Tannoudji (1994) to calculate the associated accumulated phase, which is given by integrating the classical action and dividing by Planck’s constant:

Oscillation in a harmonic potential for an integer number of trap periods produces no net phase:

Anharmonicities produce phases which can, in some cases, be found analytically Leonard and Sackett (2012) or computed by direct integration of Eq. 27.

The phase difference between two wavepackets, $\delta\phi$, is the critical quantity of interest in any interferometer; interferometric study of a particular interaction Hamiltonian is typically achieved by measuring the phase difference that it produces. Any additional phase difference arising from other Hamiltonian terms should be considered as a systematic effect and is the first metric that I use to quantify interferometer performance. In particular, it is less important that the phase difference be exactly zero, but rather that it is insensitive to experimental parameters not of interest. One such parameter of obvious importance is time — if the phase difference changes rapidly in time, the interferometer phase will be sensitive to both the precision of the experimental timing and the stability of the oscillation period of the confining potential. As such, I define the following as a metric of interferometer stability:

$\xi$ is evaluated at the point of recombination of the wavepackets, where the visibility is highest. This brings us to a second important measure of interferometer performance.

Interferometer visibility is a measure of the change in signal (for a TMI, the change in $P_{\uparrow}$) as a function of $\delta\phi$ and the measurement sensitivity is proportional to the visibility. Optimal visibility is achieved when the wavepackets are perfectly overlapped in phase-space at recombination. An expression for the visibility in the case of a 1D interferometer described above can be derived by evaluating Eq. 10. For clarity, I will insert the wavepacket-dependent interferometer phase, $\pm\phi/2$ ¹¹1The origin of this phase is unspecified and will depend on the exact details of the interferometer. It may derive from a Hamiltonian term involving the internal states, which I do not consider here.. This is in addition to the phases $\phi_{\uparrow,\downarrow}$ arising from the displacements intrinsic to the interferometer sequence.

The interferometer signal is found by making a projective measurement onto $\ket{\uparrow}$ and averaging over a thermal distribution. Doing so gives the following expression:

where I have defined $\delta\equiv\delta_{\downarrow}-\delta_{\uparrow}$ and $\theta$ is an additional phase prescribed by $\delta$. The visibility can be quickly identified as

This treatment can be easily extended to the 2D case, where each axis has an associated $\bar{n}$ and $\delta$ which in general differ between axes. The two axes can be treated independently and the associated visibilities multiplied together (equivalent to adding $\delta_{x}$ and $\delta_{y}$ in quadrature to give the separation of the wavepackets in the 4D phase space associated with the two axes).

Aside from the maximum visibility of the interferometer, it is again interesting to note sensitivity to experimental parameters such as the trap frequency. At recombination, the visibility varies in time with a Gaussian temporal profile. I will use the corresponding standard deviation, $\sigma_{\mathcal{V}}$, in units of the trap period, as a measure of sensitivity to the trap frequency.

One final metric that I shall consider is specific to the case of a Sagnac interferometer. Rotation of the interferometer produces a phase on each wavepacket,

where $m$ is the particle mass, $\vec{\mathcal{A}}$ is the enclosed area and $\vec{\Omega}$ is the rotation rate. I will assume that $\vec{\mathcal{A}}$ is aligned with $\vec{\Omega}$. The interferometer phase is proportional to the vector difference of the enclosed areas, $\delta\vec{\mathcal{A}}$. I assume that the areas enclosed by the wavepackets are opposite so $\delta\vec{\mathcal{A}}$ is equivalent to the scalar sum of the areas. Thus the enclosed area (often expressed, together with particle energy, as a ‘scale factor’) characterises the sensitivity of a Sagnac interferometer and any systematic that changes $\delta\vec{\mathcal{A}}$ is important to consider as it could jeopardize precision and accuracy.

The enclosed area for each wavepacket is readily calculated from the trajectories according to Green’s theorem:

where $\vec{L}$ is the wavepacket angular momentum. In general, this is calculated numerically, but for central potentials we can write the difference in enclosed areas as (cf. Fig. 1)

where $t$ is the oscillation period and $\delta{\vec{L}}$ is the difference in angular momentum of the wavepackets.

In the ideal case it is straightforward to calculate all the performance metrics. The phase difference, $\delta\phi$, is zero at all times by symmetry. The visibility, $\mathcal{V}$, is 1 at recombination. The value of $\delta\vec{\mathcal{A}}$ is provided by Eq. 37 and the sensitivity to experimental imperfections is clear.

Close to recombination, the wavepackets are separated in real space along one axis ²²2Recall that a second SDK removes the initial momentum so that the wavepackets are not significantly separated along either axis in velocity space. It is then straightforward to write down the time-evolution of the visibility. Using Eq. 26, the wavepacket trajectories and the small-angle approximation gives the coherent state at a time $t$ away from recombination:

where ‘$\pm$’ refers to the two wavepackets. The separation in phase space, $\delta$, is given by twice this amount. We can use Eq. 34 to write

The temporal width of the visibility peak is then

A change in the oscillation frequency, ${\rm d}\omega$, will cause inaccurate recombination and a reduction in visibility,

As an example, in both scenarios A) and B), ${\rm d}\omega/\omega=10^{-5}$ produces ${\rm d}\mathcal{V}{\sim}10^{-5}$ $({\sim}10^{-3})$ for $\bar{n}=0$ $(\bar{n}=124)$. To reduce the visibility to 10 %, ${\rm d}\omega/\omega{\sim}10^{-3}$ $({\sim}10^{-4})$ is required for $\bar{n}=0$ $(\bar{n}=124)$.

Note that the above analysis is for a single orbit; for multiple orbits, ${\rm d}\omega$ is multiplied by the number of orbits.

These imperfections are illustrated in Fig. 2b)–d). They incorporate misalignment of the directions of the momentum kicks imparted to split the wavepacket and of the initial real-space displacement of the wavepacket ³³3I note that in many experimental schemes $\theta_{\rm rel}$ is unlikely to occur, but include it for sake of generality.. With these imperfections, the trajectories are ellipses which are described by Eqs. 18–24.

With any of these imperfections, $\delta\phi$, $\mathcal{C}$ and $\sigma_{\mathcal{C}}$ are unaffected, but the enclosed areas are modified to give (cf. 37)

Either $\theta_{\rm com}$ or $\theta_{\rm disp}$ breaks the symmetry of the wavepacket trajectories, which produces a phase difference during the oscillations, however it takes the form $\delta\phi(t)\sim\sin^{2}(\omega t)$ such that $\xi=0$ at recombination and $\delta\phi$ is insensitive to changes in the trapping frequency to first order.

This imperfection is illustrated in Fig. 2e). As with the previous systematic effects, the visibility and accumulated phase are unaffected. The area enclosed by each wavepacket is modified according to Eq. 37, but $\delta\vec{\mathcal{A}}$ remains unchanged.

This experimental imperfection is not completely benign, however; a common initial velocity along $\pm\hat{x}$ makes the phase difference at recombination, while equal to zero, sensitive to the trap frequency. Using Eqs. 27 and 18–24, the phase difference is given by

where $p_{\rm SDK}=mv_{\rm SDK}$. The time derivative at $t=2\pi/\omega$ (equivalently at $t=0$) can be computed directly from this formula and is non-zero. Alternatively, one can note that

For a change in trap frequency of ${\rm d}\omega$, the change in the phase difference is

It should be noted that this error is smaller for higher trap frequencies and increases with $p_{\rm SDK}$ (which is desired for greater interferometric sensitivity). Assuming $v_{x,{\rm com}}$ equal to $v_{\rm rms}$ for scenarios A) and B) and ${\rm d}\omega/\omega=10^{-5}$ produces a phase difference of $\delta\phi\sim\pi/10$. Again, this treatment assumes a single orbit; the accumulated phase difference is multiplied by the number of orbits.

In practice, $v_{x,{\rm com}}$, and hence $\xi$, will likely have a range of values symmetrically distributed around zero. In this case, the presence of ${\rm d}\omega$ will not affect the mean value of $\delta\phi$, but will, upon taking an ensemble average, reduce the visibility, and hence the precision with which $\delta\phi$ can be measured.

This imperfection is illustrated in Fig. 2f). Note that I am assuming that the principal axes of the trap are aligned with the cartesian axes. Rotation of these axes relative to each other changes the effects of some imperfections. In particular, a trap frequency mismatch takes on the character of other imperfections, most intuitively a coupling between the axes (Sec. III.1.6).

Again by symmetry neither $\delta\phi$ nor $\mathcal{C}$ are affected by a trap frequency mismatch. However, $\delta\vec{\mathcal{A}}$ is affected as the confining potential is no longer central. Let $\omega_{y}$ change by an amount ${\rm d}\omega$. Using Eq. 36 the enclosed area is, to first order in ${\rm d}\omega$,

It should also be noted that the trajectories are no longer closed. In particular, for a single orbit, a displacement of

relative to the initial position is produced. Such a shift is unlikely to be problematic, e.g. in scenario A) a $10^{-5}$ mismatch between frequencies produces a negligible $0.1$ pm of displacement per orbit. Of course, significantly larger trap instabilities should be avoided.

This imperfection is illustrated in Fig. 2g) and consists of modifying the trapping potential (Eq. 17) to contain terms proportional to $x^{4}$, $y^{4}$ and $x^{2}y^{2}$. I ignore terms like $V{\sim}xy^{3},x^{3}y$ as these are generally negligible due to the symmetry of the apparatus used to generate the trapping potential. The only metric impacted is the enclosed area. The resulting trajectories are most similar to the case with $\omega_{x}\neq\omega_{y}$ (Sec. III.1.4) and suffer from the same problem that the recombination is offset from the initial position.

This imperfection is illustrated in Fig. 2h). It is a modification of the trapping potential to include coupling between the two spatial axes. This is by far the most detrimental imperfection considered here and the only one that causes the orbits not to close. As such, multiple metrics of the interferometer are affected. Unlike the other systematics considered, the symmetry of the trajectories is broken in both $x$ and $y$, meaning that the recombination is affected.