1.2.1 Surpassing SQL with spin-squeezed state
We resume from section 1.1 and consider the quantum uncertainties in the collective spin in a theoretical setting. A Heisenberg uncertainty relation between the spin operators directly results from the commutation relations (Eq. 1.2):
∆Sn∆S⊥≥ |h ˆSsi|/2 (1.23)
where n is an axis perpendicular to the mean spin direction s, and ⊥ is orthogonal to both n and s. ∆Sn = qh ˆSn2i − h ˆSni2 denotes the standard deviation of an ensemble of measurements of ˆSn.
For the CSS, |θ, φ, Ni, a projective measurement along any direction ⊥ orthogonal to s will project each qubit into the two eigenstates with equal probability. Therefore h ˆS⊥i = 0
and the variance of the measurement (∆S⊥)2 = N/4, coming from the sum of all qubits.
This can be seen for example from
(∆Sz)2 = 1
It can be shown that the CSS is the optimal separable state for metrology [25], as it saturates the Heisenberg inequality. It exhibits an equal distribution of the uncertainty
∆Sn = ∆S⊥ = √
N /2. This isotropic minimum uncertainty is known as the standard quantum limit (SQL), which leads to the QPN in the clock measurements. In terms of a phase measurement, we are interested in the angular resolution on the Bloch sphere:
∆θSQL= ∆S⊥,CSS
|h ˆSsi| = √1
N, (1.24)
Nevertheless, the isotropic uncertainty distribution originates from the separable state, but the uncertainty relations do not prevent a single projection from being more precise than the SQL. It turns out that entanglement between particles can lead to such “redistribution”
of the uncertainty, resulting in a “squeezed” state, namely ∆S⊥<√
N /2. Using the squeezed quadrature therefore enables measurement precisions beyond the SQL.
Certainly, many other entangled states exhibit metrological gain, generally described using Quantum Fisher information (see e.g. [25]), which is beyond the scope of this thesis.
Squeezing measures Without knowing the details of the entanglement, spin squeezing can be simply assessed by the standard uncertainty. The minimum uncertainty normalised to the SQL,
is sometimes referred to as the number squeezing if ξN2 <1 [28]. It is more metrologically relevant to assess the angular resolution ∆θ, in which the length of the spin |h ˆSsi|measures the “resource” in a practical squeezing process – taking into account the coherence, or the contrast of the Ramsey fringes. This metric, known as the Wineland criterion for spin squeezing [29], is defined as
ξ2 <1 signals a metrological spin squeezing, it is shown to be a sufficient condition of certain types of entanglement [80]. ξ−2 is commonly reported in dB, as it shows the effective gain in atom number to achieve the same resolution.
While as the noise distribution is squeezed in one axis, the other quadrature is necessarily anti-squeezed to satisfy the Heisenberg uncertainty relation (Eq. 1.23). But the latter is not always saturated. Squeezed states that saturate the uncertainty are called minimum uncertainly states or optimal squeezed states. A lower bound of ξ2 can be found for these states [25]:
ξ2≥ 2
N + 2 (1.27)
which is essentially the ultimate Heisenberg limit. For a detailed review of many other squeezing measures, see for example [81,25].
Motivations for trapped-atom clocks As we know, the SQL is not a remote theoretical limit. The state-of-the-art atomic fountains have long reached a short-term stability limited by the QPN [23]. Here I would like to emphasise that spin squeezing surpassing the SQL is particularly relevant for trapped-atom clocks:
• For trapped-atom clocks in which the collisional shift can impose a large uncertainty due to high density, the number of atoms is often limited. Employing spin-squeezed states can mitigate the severe SQL.
• In recently developed 3D lattice clocks in which collisional interactions are suppressed by keeping a single atom per lattice site [65], increasing atom number becomes techni-cally difficult and the QPN limit is expected to be approached soon.
• More generally, squeezed states can benefit compact applications where the number of atoms is limited, either technically or fundamentally.
1.2.2 Overview of spin-squeezing generation
With all these prospects, there have been tremendous experimental efforts over a decade to demonstrate and study spin squeezing in atomic systems. Here I give a brief summary of the most studied methods for squeezing generation.
As a particular type of entanglement, spin squeezing correlates the local spin observables of the atoms. Let us distinguish two categories of entanglement creation, namely by dynamics due to interactions between atoms or by a partial projection of the collective state.
1.2.2.1 Squeezing by inter-atomic interactions
Described in terms of collective spin operators, entanglement requires non-linearity. A bench-mark model is the so-called one-axis twisting (OAT) Hamiltonian, with the simplest non-linearity:
HˆOAT= ~χ ˆSz2 (1.28)
The collective operator ˆSz2 actually means that each atom interacts with all others. But it also gives a very intuitive picture: the precession rate is itself proportional to Sz, distorting the Bloch sphere (Fig. 1.3(b)). The initially isotropic noise distribution is twisted under the dynamics. It can be shown that the process almost preserves the minimum uncertainty area and exhibits squeezing along a certain axis [28]. The dynamics reaches a maximum squeezing parameter around tmax ∼ χ−1N−2/3 and later loses the squeezing as the state wraps around the Bloch sphere, but the entanglement keeps increasing, and the assessment of which requires more complex measures like the nonlinear squeezing [82]. The maximum squeezing at tmax scales with ξ2 ∼ N−2/3.
Collisional interactions in BECs As widely explored, collisional interactions in a BEC lead to an OAT Hamiltonian, in which χ depends on the scattering lengths and the overlap of the wavefunctions of the condensate modes. By controlling the scattering lengths via
Feshbach resonances [33, 36] or modifying the wavefunction overlap via state-dependent potentials [34,35], substantial spin squeezing has been achieved. However, strong interactions in a BEC generally limit their application for clocks.
Cavity feedback As will be detailed below, light-mediated interactions between atoms in an optical cavity can also produce an effective OAT Hamiltonian [43, 44]. The χ term now depends on the dispersive coupling between atoms and cavity photons, and can be very large in a strong-coupling cavity-QED system. However, there is usually excessive noise enhancement in the anti-squeezed quadrature that far exceeds the squeezing due to cavity decay. This so-called non-unitary anti-squeezing compromises possible metrological gain in clock applications [83], but recent progress has approached near unitary squeezing [84,45].
Beyond OAT A few other schemes show metrological advantages compared to the OAT, including the “twist-and-turn” [85] and two-axis-counter-twisting (TACT) [28]. The for-mer has been demonstrated in a BEC [86], while the TACT, despite extensive theoretical studies, remains experimentally challenging. Ideally, the TACT, described by a Hamiltonian HˆTACT= ~α( ˆSθ2− ˆSθ2+π/2) can achieve maximum squeezing at the Heisenberg limit ξ2 ∼1/N, and it squeezes faster compared to OAT (ξ2 ∼ N−2/3). However, the benefits can be lost in the presence of decoherence [87].
(a) (b) (c)
Figure 1.3 Conceptual representation of squeezing by QND measurement and by OAT dynamics. (a) depicts a CSS prepared on the equator with its spin uncertainty. (b) illus-trates the OAT dynamics: the precession rate depends on Sz. The twisted noise distribution exhibits minimum uncertainty along the axis of ~z rotated by α0. (c) shows the squeezing by QND measurement. h ˆSzi after each measurement depends on the measurement re-sult, which itself fluctuates as the SQL, but ∆Sz is deterministically reduced after the measurement, limited by the measurement uncertainty. The squeezing is revealed by ∆Sz
conditioned on the measurement outcome unless feedback correction is applied. In both (b) and (c), the dashed outer spheres indicate the possible coherence loss due to the squeezing process.
1.2.2.2 Squeezing by quantum non-demolition measurement
A QND measurement is a measurement that preserves the measured observable. For exam-ple, if the Hamiltonian of the measurement process satisfies [ ˆSz, ˆHQND] = 0, meaning that Sz is a constant of motion, measuring an observable in ˆHQND (that couples to ˆSz) can realise a QND measurement of ˆSz.
The measurement projects the collective spin ˆSz but not individual atoms. In fact, this partial projection leads to entanglement that can modify the noise distribution, e.g. ∆Sz. In another point of view, the collective observable ( ˆSz) is known to the limit of the measurement uncertainty, which can be more precise than the SQL.
Since the squeezed state after the measurement (e.g. Sz) depends on the measurement result, the squeezing is “conditional” if one looks at the measurement outcomes. But af-ter each measurement, ∆Sz is deterministically reduced. Unconditional squeezed can be achieved by correcting Sz after each measurement using the information of the measurement result [88,89].
QND measurements of atomic systems are generally realised by atom-light interactions.
The measurement uncertainty is then limited by the photon shot noise (PSN) of the probe light, before reaching the maximum achievable squeezing that is set by non-ideal effects such as decoherence due to photon scattering and Raman spin-flip processes.
Free-space QND measurements can be implemented in free-space, based on i) the phase shift induced by the dispersive coupling, as shown in one of the first demonstrations of spin squeezing [39]; ii) and the Faraday rotation of light [90,38,91], a promising candidate for quantum-enhanced magnetometry. Nevertheless, high level of squeezing is generally difficult due to weak coupling (optical depth) in a single-pass geometry.
Cavity-QED The optical depth of the atoms is strongly enhanced in an optical cavity as each photon travels many round-trips interacting with the atoms, entering the realm of (collective) strong coupling. Since the early demonstration of conditional spin-squeezing based on cavity measurements [41], squeezing records have been quickly updated [92,89], and reached lately almost 20 dB [42]. These include atom-light coupling both in the dispersive regime [41,42] and in the resonant regime [93,89].
We will discuss the cavity measurement scheme in more detail. The squeezing results in this thesis are conditional squeezing by cavity measurement, which belongs to this category.
QND measurements for clocks On the other hand, QND measurements in their on right are of great interest for clocks. Here even a coherence-preserving quantum non-destructive measurement can be employed to perform multiple Ramsey cycles reusing the same atoms, which will improve the duty cycle of the clock. It is particularly interesting for clocks suffering from Dick effect such as many current optical clocks [94, 95], and compact clocks (like ours). For true QND measurements, atoms are not only recycled but also stay in coherence. For example, (weak) QND measurements only lock the LO (in presence of phase noise) roughly in phase with the atoms to ensure the phase sensitivity. It has been demonstrated that QND measurements outperform uncorrelated Ramsey sequences with recycled atoms [96,56], opening ways to improving clocks and interferometer-based sensors.