Quantum metrology with Bloch Oscillations in Floquet phase space

It is well known that quantum particles confined in periodic potentials and subjected to a constant force do not accelerate uniformly in real space. Rather, they undergo Bloch oscillations, which have been observed in a variety of systems, including ultracold atom gases trapped on optical lattices Dahan1996 ; Wilkinson1996 ; Gustavsson2008 and optical waves in waveguide arrays Pertsch1999 ; Morandotti1999 . Since the Bloch oscillations frequency depends only on the lattice spacing and the applied external force it provides an accurate tool for the measurement of weak forces and the precise determination of fundamental constants Poli2011 ; Biraben2004 ; Biraben2006 ; Biraben2011 .

There is much interest in extending the study of these systems to situations where they are subjected in addition to a time-periodic forcing Arlinghaus2011 ; Kudo2011 . This includes in particular the study of ‘super-Bloch oscillations’, which can result in linear transport in a lattice Haller2010 and spatial oscillations of the system even in the absence of a spatial lattice Junk2020 . More generally, time-periodic forcing is now exploited in the emerging area of ‘Floquet engineering’, a powerful tool to control and modify quantum systems, including for instance the discovery of new out-of-equilibrium phases, see e.g. Moessner2017 ; Weitenberg2021 , in particular, the Floquet time-crystal phase Else2016 ; Sacha2018a . Applications of these techniques to enhance the precision of quantum metrology have also been proposed, see e.g. Lukin2020 ; Fiderer2018 ; Jiang2021 .

The present work extends the Bloch measurement idea to the Bloch-like oscillations that appear in the Floquet phase space of a system subjected to a periodic temporal drive rather than a spatially periodic potential. Just like in the case of Bloch oscillations, the measurement of constant forces applied to systems undergoing such oscillations reduces to frequency measurements, with the remarkable precision with which they can be carried out. But as we shall show, the transformation of the Hamiltonian of the ‘measuring apparatus’ to the phase representation brings to the fore two key features of this approach, as compared to the familiar Bloch oscillations approach: It is not limited to the measurement of constant forces, and it applies to a variety of physical observables. We will illustrate these points on the specific examples of the realization of a tachometer and a magnetometer.

The paper is organized as follows: Section II presents the analytical analysis of the underlying idea under a secular approximation, which is then revisited via numerical simulations in Section III. Section IV then applies our proposed scheme approach to the examples of a tachometer and a magnetometer. Finally, Section V is a summary and outlook. Some additional details of the relevant action-angle transformations are given in an Appendix.

We consider a system described by the one-dimensional Hamiltonian

where $U(x)$ is a conservative and non-periodic potential and $H_{1}=V(x)\cos\omega t$ is a weak single-frequency perturbation oscillating at the frequency $\omega$.

Two common strategies can be used to reveal the lattice structure of this system in Floquet phase space. The first one is based on a perturbative approach combined with a specific time-dependent canonical transformation Berman1977 ; Berman1981 ; Berman1985 ; Buchleitner2002 , while the second one relies on the use of Floquet quantum states and quasi-energy theory Grifoni1998 ; Holthaus2016 . We adopt the first method here, as it is particularly convenient to discuss the various measurement applications that we have in mind.

Before illustrating this measurement technique on several concrete examples we first need to revisit the assumptions and limitations underlying Eq. (20), as they inform the range of experimental situations achievable in practice. In particular, we have seen that the amplitude of the Bloch oscillations is exponentially suppressed by the lattice depth. This feature needs to be reconciled with our assumption that only the first band of the system is initially populated, which requires that the band gap, roughly equal to $V_{n}$, be much larger than $V_{\rm probe}$, which scales as $(2\pi/n)f$. In addition, the secular approximation holds only under the perturbation condition $V_{n}\ll\hbar\Omega$, since otherwise the off-resonant driving becomes significant and can result in irregular dynamics.

To address these issues more quantitatively we have carried out detailed numerical simulations for the case of a particle of mass $m$ trapped in an infinite square well $U(x)$ of width $L$ and subjected to a single-frequency periodic driving at frequency $\omega$ and a sequence of $\delta$-kicks in alternating directions, with Hamiltonian

For this square potential, the action-angle transformation between the position $x$ and $\Theta$ is linear,

and the angular frequency is $\Omega=\sqrt{2\pi^{2}E_{0}/mL^{2}}$ with $E_{0}$ the initial energy, see the Appendix for the details of the transformation. For a Gaussian wave packet with a phase factor $e^{ip_{0}x/\hbar}$, which corresponds to a classical particle with initial kinetic energy $E_{0}=p_{0}^{2}/2m$, the angular frequency becomes $\Omega=\pi p_{0}/mL$, the oscillation frequency of a wave packet in a square well of width $L$.

As derived in the previous section, in the Floquet phase space the periodic driving is resolved into multiple components as a Fourier series in the angle $\Theta$,

so that, with the integral (5) and the transform (23), the only non-vanishing amplitudes are for $\ell$ odd, $V_{\ell}=2gL/\pi^{2}\ell^{2}$. Under the secular approximation only the components $\ell=\pm n$ survive, resulting in an effective $n$-site lattice potential, $(2gL/\pi^{2}n^{2})\cos n\vartheta$.

Similarly, in the Floquet phase space, the sequence of $\delta$-kicks is resolved into multiple components

where the kick duration $\tau\ll T_{\rm D}$. The argument leading to Eq. (21) shows that under the secular approximation, they act effectively as the linear potential $f\vartheta$ due to a constant weak force $f=LF_{0}\tau/\pi T_{\rm D}$, resulting in the Floquet-Bloch oscillations with the period $T_{\rm B}=\hbar n\pi T_{\rm D}/LF_{0}\tau$. Then the amplitude of the kick force can be determined by direct measurement of the frequency of the Floquet-Bloch oscillations as

In our simulation, we used the original Hamiltonian (22) in coordinate space instead of the perturbative Floquet space lattice Hamiltonian obtained in the secular approximation. We also accounted for the decoherence induced by thermal motion by describing the evolution of the quantum state of the particle via the master equation with quantum Brownian noise Gardiner1999 ; Caldeira1983

with $\gamma$ and $\beta$ the damping rate and the inverse temperature, respectively.

The results of the simulations with a fixed value of $n=\Omega/\omega$ and for several values $V_{n}$ are summarized in Fig. 1, which compares the time evolution of the mean position $\langle x(t)\rangle$ without (left red plots) and with (right blue plots) the secular approximation. For $V_{n}\ll\hbar\Omega$, the two solutions almost overlap, confirming the validity of the secular approximation in that regime. However, larger values of $V_{n}$ result in an increased discrepancy between the exact and approximate results, confirming that the approximation only holds provided that $H_{0}\gg H_{1}$. In particular, for $V_{n}$ close to $\hbar\Omega$, $\langle\hat{x}(t)\rangle$ exhibits a disordered evolution completely absent from the secular approximation behavior.

Fig. 1 also illustrates that the Floquet-Bloch oscillations evident in the periodic changes of the envelope of $\langle x(t)\rangle$ are only visible for intermediate values of $V_{n}$. More specifically, in the top figure ($V_{n}=0.0009\hbar\Omega$) we have that $V_{n}\lesssim V_{\rm probe}$ and the assumption underlying our analysis that $V_{\rm probe}$ does not induce interband transitions ceases to hold, even though the secular approximation works well. This shows that the proposed measurement scheme requires that $V_{n}$ satisfies the ‘Goldilocks condition’ of being large enough that no interband transition takes place, but small enough that the secular approximation holds approximatively.

The effects of dissipation are shown in Fig. 2, which compares $\langle x(t)\rangle$ in the presence of dissipation resulting from coupling to a Brownian reservoir to the case without dissipation. In these plots, $V_{n}$ is kept constant, but the frequency ratio $n=\Omega/\omega$ takes increasing values $n=$ 15, 21 and 31 from the upper to the lower plot, so as to illustrate both the dependence of the oscillation frequency on $n$ and the effects of the scaling of $V_{\rm probe}$ on $(2\pi/n)f$.

In the absence of dissipation, both the period and the amplitude of the Floquet-Bloch oscillations increase with increasing $n$, consistently with the expression (18) for the oscillation period and with Eq. (19) which shows that the decrease of the normalized lattice depth $s$ for increasing $n$ results in an increase of the width of the first band, and hence an increase in the amplitude of oscillations. In Fourier space the amplitude of the Bloch frequency peaks at $\Omega\pm 2\pi/T_{\rm B}$ become therefore increasingly visible. Note also that for $n=15$, the top part of Fig. 2, the oscillations in $\langle x(t)\rangle$ are strongly perturbed, a result of the appearance of interband transitions for the relatively large $V_{\rm probe}$ associated with this small $n$.

Turning now to the effect of dissipation, we observe that the damping of $\langle x(t)\rangle$ becomes faster as $n$ is increased. That is because the noise broadens the frequency spectrum of the system. For increased $n$ a correspondingly larger number of non-resonant low-frequency modes with $|\ell|<n$, which are ignored in the secular approximation, come into play. While they all have the same damping rate $\gamma$ for Brownian noise, their coupling to the resonant mode results in an increased decoherence rate $\kappa$ of the Floquet-Bloch oscillations. From our numerical simulations we estimate it scales approximately with $n$, $\kappa\sim n\gamma$. As a result, for increasing $n$ the frequency peaks first broaden and eventually disappear.

Combined with the results summarized in Fig. 1, these results illustrate that the frequency peaks of the Floquet-Bloch oscillations in the displacement spectrum are only distinct enough to determine $f$, and hence $F_{0}$, when the system parameters satisfy the inequalities

With these limitations in mind, we now turn to a secular approximation discussion of several potentials $U(x)$ appropriate for the Floquet-Bloch measurement of a variety of observables.

We now turn a brief discussion of Floquet-Bloch measurement schemes based on other potentials $U(x)$ and demonstrate how they can be exploited, for instance, as tachometers or magnetometers.

We consider first the one-dimensional triangular well potential

in which case the position and momentum $x$ and $p$ are related to the action-angle variables by Buchleitner2002

where the angular frequency is $\Omega=\eta\pi/\sqrt{2mE_{0}}$, see the Appendix. Importantly, it is now the momentum that is linear in $\Theta$, suggesting that this potential might find applications in situations where $V_{\rm probe}$ depends linearly on the momentum $p$ rather than $x$. We present two examples that exploit such a situation, the first one resulting in the realization of a tachometer as shown in Fig. 3, and the second in a magnetometer.

In this work, we have exploited Floquet engineering techniques to extend the concept of Bloch-oscillations-based precision measurements to the use of Floquet-Bloch oscillations. Specifically, we showed that for a quantum system trapped by some potential $U({\bf r})$, the addition of a weak single-frequency perturbation, combined with a sequence of $\delta$-kicks, results in the onset of the Bloch-like oscillations in its Floquet phase space. These oscillations present significant advantages and flexibility over traditional Bloch oscillations for precision measurements, as a proper design of $U({\bf r})$ and kicking interactions permit to exploit them for the measurement of a broad variety of observables, including rotation rates, magnetic fields, and even the strengths of singular potentials.

Our proposed scheme requires however the efficient continuous measurement of mean positions or momenta to determine the precise frequency of the Floquet-Bloch oscillations, and the restriction of our analysis to the first quasienergy band implies in addition that this approach is limited to the detection of weak signals. For signals strong enough to excite interband transitions, a measurement of the emission spectrum of the Floquet system is possible, see e.g. Ref. Jiang2022 , and may be more advantageous. A quantitative comparison of the relative benefits and limitations of both approaches will be the subject of future work.

The present paper concentrated on single-particle physics. Floquet-Bloch oscillations are vulnerable to dissipation-induced decoherence, and this imposes as we have seen limitations on the range of accessible Floquet-Bloch frequencies. A possible way to circumvent this problem might involve replacing the system with Floquet time crystals Else2016 , which can withstand dissipative environments Gong2018 ; Zhu2019 ; Hemmerich2021 . We will show in future work how measurement schemes similar to those discussed here can be developed in such systems by using appropriately modulated many-body interactions.

If $H_{0}$ dominates over all other terms in the Hamiltonian and $H_{1,2}$ can be ignored, the motion of the system is periodic, and the action-angle transform is realized by integration over one period of the motion,

where $E_{0}$ is the energy of the system and $x_{\max,\min}$ represent two peak positions of the periodic motion in the physical coordinates. The angular velocity is given by

and the relation between $\Theta$ and the physical coordinates is given by the integral

where $\Theta_{0}$ and $x_{0}$ represent their initial values.

For example, if the conservative potential in $H_{0}$ is a one-dimensional infinite square well,

from Eqs. (11, 53, 54) we obtain

The integral (55) gives

Further taking into account that $x\in\left[0,L\right]$ and $\Theta\in\left[0,2\pi\right)$, we then obtain

which also gives

Similarly, if the conservative potential is a one-dimensional triangular well

we obtain

and then the transforms,

As a final example, consider the infinite one-dimensional well with a quartic bottom profile

with the boundary $L\gg b/\pi>0$. For a small enough initial energy, $|E_{0}|\ll m\Omega_{0}^{2}b^{2}/(2\pi^{4})$, we find the approximate expressions,

and then the approximate transforms,

It follows that in the case of a probe potential of the form $V_{\rm probe}(x)=-a/x$, the inverse $x$-dependence results in a linear angular dependence in the phase space,

If the probe potential is modulated by a sequence of $\delta$-kicks in alternating directions,

in the Floquet phase space, under the secular approximation (21), they act as a linear potential $f\vartheta$ with the force,

So the potential strength $a$ can be obtained from the Bloch period as