Experimental measurement of quantum-first-passage-time distributions

TL;DR

This paper reports the first experimental measurement of quantum first-passage-time distributions using a trapped ion system, introducing a novel measurement protocol applicable to various quantum systems.

Contribution

It develops a new composite-phase laser pulse sequence for tunable measurements of quantum first-passage times in a trapped ion.

Findings

Measured QFPTDs of ion energy under electric-field noise

Established connection between quantum and classical FPTDs

Developed a broadly applicable measurement protocol

Abstract

Classical First-Passage-Time Distributions (FPTDs) have been extensively studied both theoretically and experimentally. Their quantum counterparts, Quantum First-Passage-Time Distributions (QFPTDs), remain largely unexplored and have deep implications for both fundamental physics and the development of emerging quantum technologies. We measure the first QFPTDs using a motional mode of a single trapped ion. We develop a novel composite-phase laser pulse sequence to perform tunable stroboscopic projective measurements of the motional state of a trapped ion. We measure QFPTDs of the ion energy when coupled to electric-field noise and establish a clear connection with its classical counterpart. The measurement protocol developed here is broadly applicable to other quantum systems and provides a powerful method for exploring a broad range of QFPTD phenomena. With these results we open a new field of experimental investigations of QFPT processes with potential future relevance to quantum search algorithms, unraveling connections between classical and quantum dynamics, and study of the quantum measurement problem.

Equations

$$P^{\text{FPT}}(n\theta )=\prod _{i=0}^{n-1}{P}^{\text{S}}(i\theta )\cdot P^{\text{A}}(n\theta )$$

$$\dot{\rho }=\frac{⟨\dot{n}⟩}{2}(2a\rho a^{†}-a^{†}a\rho -\rho a^{†}a+2a^{†}\rho a-a{a}^{†}\rho -\rho a{a}^{†})$$

$$|{\psi }_0⟩=|D_{5/2}⟩\otimes |0⟩.$$

$$\ket{\psi(\theta^{-})}=\ket{D_{5/2}}\otimes\bigg{(}\sum_{n=0}^{\infty}d_{n}(\theta)\ket{n}\bigg{)}.$$

$$\begin{split}\ket{\psi(\theta^{+})}&=\ket{D_{5/2}}\otimes\bigg{(}\sum_{n=0}^{N_{B}-1}d_{n}(\theta)\ket{n}\bigg{)}\\ &\hskip 30.0pt+\ket{S_{1/2}}\otimes\bigg{(}\sum_{N_{B}}^{\infty}d_{n}(\theta)\ket{n}\bigg{)}.\\ \end{split}$$

Full text

††preprint: APS/123-QED

Introduction—The first-passage time is typically defined as the time that a system dynamical observable is first measured to be outside a surviving domain. Distributions of these times, known as First-Passage-Time Distributions (FPTDs), have a long history in science. They reveal individual trajectory dynamics that ensemble quantities fail to capture [1, 2, 3], as exemplified by Schrödinger’s clarification of Millikan’s oil drop experiment [4]. FPTDs have since been used in diverse areas such as in the study of activation in chemical and biological processes [3, 5, 6, 7, 8], current-switching in electronic transport structures [9, 10], economics and market models [11, 12, 13, 14], and climate science [15, 16, 17].

The development of quantum walk search [18, 19, 20, 21, 22]—prompted by advances in quantum-state manipulation—has given rise to Quantum-First-Passage-Time Distributions (QFPTDs) as a natural way to express the search problem [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. There are important differences to consider between the classical and the quantum FPTD problem. Whereas in the classical case FPTDs can only be defined for stochastic processes (a deterministic process would have a delta-function FPTD), in the quantum case, measurements themselves introduce randomness. Thus, even a unitarily evolving quantum system with interspersed measurements has a non-trivial distribution of first-passage times. Additional quantum properties such as the superposition principle and the related (anti-)Zeno effect—the (enhancement) suppression of quantum evolution by measurement [37]—make QFPTDs richer than their classical counterpart, and place them at the interface of quantum and classical mechanics. Fig. 1 presents sample trajectories that illustrate key conceptual differences between quantum and classical FPTDs.

Measurements at regular intervals (stroboscopic) have become an accepted method of interpreting the problem of QFPTDs [26], although measurements at random intervals [28] and continuous measurements of open quantum systems [36] have been investigated. Furthermore, since time is not a self-adjoint operator, QFPTDs are ambiguously defined in the continuous measurement limit [25, 38, 39]. This ambiguity highlights the need for experimental validation of newly obtained theoretical results—among them, the exotic predictions of spin-dependent arrival times [40, 41, 42]. The development of experimental QFPTD methods has potential to further our understanding of analytically challenging quantum search algorithms with known exponential speedup and support their possible implementation [24]. We expect that QFPTDs can be used for precision measurement as repeatedly monitored quantum systems offer sensing advantages, notably through quantum hindsight effects [43, 44, 45]. Trapped-atomic ions are an ideal platform for experimental QFPTD measurements. Their prominence in quantum information processing has led to the development of a variety of coherent [46] and incoherent [47] control techniques that can be leveraged to simulate a broad variety of QFPTDs.

In this letter we develop and implement a novel method to experimentally perform the first QFPTD measurement through use of the quantized motional state of a harmonically trapped ion. The first-passage time is defined as the first time the ion is measured to have an energy greater than or equal to a certain threshold $E_{B}=\hbar\omega(N_{B}+1/2)$ after initialization in the ground state, under stroboscopic measurement. The surviving domain is defined in terms of the energy eigenstates $\ket{n}$ of the unperturbed oscillator: $\{\ket{0},\ket{1},\ket{2},...,\ket{N_{B}-1}\}$. The classical analogue of this QFPTD is non-trivial and subtle due to the multi-dimensional boundary condition [48]. We define a ‘passage measurement’ (experimental implementation described below) with two possible outcomes (‘survival’ and ‘absorption’), whose projectors $\mathbb{P}^{\text{S}}=\sum_{n=0}^{N_{B}-1}\ket{n}\bra{n}$ and $\mathbb{P}^{\text{A}}=\sum_{n=N_{B}}^{\infty}\ket{n}\bra{n}$ correspond to the surviving and absorbing domains, respectively. We dub the measurement ‘step pulse’ as it effectively forms a quantum-low-pass filter for the energy eigenstates, which is seen clearly in Fig. 2b. Since measurements alter the wavefunction, even measurements that do not signal a passage can modify the trajectory—a fundamentally quantum effect. We note that the step pulse measurement decreases the energy of the state in all cases where a survival outcome is obtained. For stroboscopic measurements with fixed interval $\theta$, the QFPTD is

[TABLE]

where $P^{\text{S}(\text{A})}(i\theta)$ is the probability of obtaining a survival (absorption) outcome from the $i^{\text{th}}$ measurement, given that all prior measurements resulted in survival. The first passage occurs when the first absorption measurement is obtained.

Theoretical background—In quantum mechanics, we distinguish between closed systems, which evolve unitarily, and open systems, which do not. The QFPTD measurement scheme developed in this letter can be used in both cases. In this experiment, the ion is coupled to the natural, high-temperature amplitude reservoir from noisy electric fields in the environment $\xi(t)$ [47]. The electric-field noise causes an initially ground-state cooled ion to heat and evolve to a thermal state of average occupation number $\bar{n}$. We independently measure the heating rate $\dot{\bar{n}}$ at the motional frequency $\omega$ through the standard sideband asymmetry technique. We then relate the heating rate to the noise spectral density using $\dot{\bar{n}}(\omega)=(e^{2}/4m\omega\hbar)S_{\xi}(\omega)$ [49]. The time evolution of this process, from which we calculate the QFPTD, can be described using either quantum trajectory theory [50], or a master equation for the density matrix $\rho$ [47]

[TABLE]

where $a^{\dagger}(a)$ is the motional state creation (annihilation) operator. While both approaches yield the same results, they differ in terms of computational efficiency and interpretability. This evolution closely resembles a well-known classical analogue: the classical undamped harmonic oscillator driven by additive noise: $\ddot{x}+\omega^{2}x=e\xi(t)/m$ where $m$ is the mass of the ion. For convenience, we henceforth adopt dimensionless time $t$, such that $\dot{\bar{n}}=1$.

We note several features of the theoretically calculated QFPTD, which we can solve exactly in the $\theta\mathord{\to}0$ limit. For finite $\theta$, we predict a reduction of the mean first-passage time for decreasing $\theta$. This effect is reflected in the escape probability $E(t;\theta)=\sum_{i=1}^{t/\theta}P^{\text{FPT}}(i\theta)$ at time $t$, which is larger for smaller $\theta$: i.e. $E(t;\theta_{1})<E(t;\theta_{2})$ if $\theta_{1}>\theta_{2}$. This enhancement is shown in Fig. 4. Although it appears to be an anti-Zeno effect, the enhancement of the escape probability arises because faster probing detects an escape sooner. This process is analogous to evaporative cooling of atom clouds (in particular to the RF knife technique [51]), where in this case, surviving trajectories have lower average energy than if non-selective measurements had been performed. Due to the quantization of energy, if $N_{B}=1$, the QFPTD is a pure exponential without an initial ballistic part, as each surviving measurement projects the energy back into the ground state, and thus the trajectory begins in its statistical steady state. However, for $N_{B}\geq 2$ the statistical steady state is only achieved around $t\gtrsim N_{B}$, at which point the QFPTD transitions from the ballistic to the diffusive regime. This is shown in the Supplementary Material (SM).

Despite the energy range ($\sim\hbar\omega$) of our experiment being quantized and the measurement protocol implementing projective measurement, the resulting QFPTD is similar (for $N_{B}\geq 2$) to the equivalent classical FPTD whose moments may obtained using stochastic averaging in the $\theta\mathord{\to}0$ limit [52, 53]. Namely, the long-time tail of the QFPTD is exponential, $\lim_{t\rightarrow\infty}P^{\text{FPT}}(t)\sim e^{-\beta t}$, where $\beta$ scales linearly with the $E_{B}$ [54]. This can be seen in Fig. 3. Furthermore, the first- and second-moments of the quantum and classical FPTDs agree (see the SM for more details).

Experimental implementation—We use a novel composite-phase laser pulse sequence to perform the step pulse measurement on a trapped 40Ca+ ion, whose internal states are shown in Fig. 2c. Composite pulses, first used in NMR spectroscopy, have been used in a variety of contexts to cancel errors (such as the famous SK1 and BB1 sequences [55, 56, 57]), and more recently to detect occupation of harmonic oscillator states [58]. Detuning a $729$ nm laser by the motional frequency above the $\ket{S_{1/2}}\leftrightarrow\ket{D_{5/2}}$ transition gives rise to an anti-Jaynes-Cummings Hamiltonian. This interaction, known as the first blue sideband (BSB), couples the $\ket{S_{1/2}}\otimes\ket{n}$ and $\ket{D_{5/2}}\otimes\ket{n+1}$ states with $n$-dependent coupling strength $\Omega_{n,n+1}$ [46]. We exploit the coupling-strength dependence on $n$ to perform an effective $\pi-$pulse on the internal state if $\ket{n\geq N_{B}}$ using a series of pulses of different relative phases and durations, which we find using numerical optimization. The pulse sequences used in this experiment are shown in Fig. 2a. We also show a plot of the expected excitation probability as a function of the motional quantum number $n$ for different $N_{B}$ in Fig. 2b, from which the step-like nature of the pulse is clear. In principle, with pulses of sufficient duration and complexity, this conditional excitation can be performed with near-unit fidelity. However, due to the limited coherence time of the ion and control errors, there is, in practice, a trade-off between the increase in fidelity gained by increasing the duration (and complexity) of the composite pulse, and the effects of decoherence which get worse with increasing time. See the SM for more details on the step pulse and these trade-offs.

In order to measure the QFPTD, we perform repeated trials and aggregate the first-passage times. A single trial proceeds as follows: the ion is ground state cooled using resolved-sideband cooling [46], and the internal state is pumped to $\ket{D_{5/2}}$. Thus each trial starts with the ion in

[TABLE]

Then, in the quantum trajectories picture, as the ion heats due to the electric-field noise, the motional state amplitudes $d_{n}$ evolve stochastically for a time $\theta$ [50]

[TABLE]

We then apply the composite-phase pulse that flips the internal state if the motional state is $\ket{n\geq N_{B}}$, and leaves the internal state unaffected if the motional state is $\ket{n<N_{B}}$

[TABLE]

After the composite pulse, the ion is in an entangled state between internal energy levels and the motion. A subsequent measurement of the ion internal state using state-dependent resonance fluorescence has two possible outcomes (Fig. 2c): $\ket{S_{1/2}}$ in which case the ion fluoresces and photons are rapidly emitted (so-called ‘bright’ state), or it can collapse to the $\ket{D_{5/2}}$ state and emit no photons (so-called ‘dark’ state). Camera images of the ion in both bright and dark states are shown in Fig. 2b. This measurement of the internal state implements the measurement projectors $\mathbb{P}^{\text{S}}$ and $\mathbb{P}^{\text{A}}$. If a survival outcome is obtained, the trial continues for another time interval $\theta$ and the step pulse is re-applied. This sequence is repeated until a bright measurement is obtained, after which the trial is terminated.

We have tacitly assumed that there is no spontaneous emission of the $\ket{D_{5/2}}$ state (lifetime 1.2s [59]) between stroboscopic measurements. The duration of these experiments is sufficiently short that errors due to spontaneous emission do not dominate. While this poses a limitation to directly measuring QFPTDs which go out to long times, this problem can be circumvented by using either longer-lived optical qubits, such as Barium [60], or ground-state qubits with practically infinite lifetimes [61]. Furthermore, given knowledge of the excited state lifetime, the true QFPTD can be reconstructed from the observed QFPTD which has been modified by spontaneous emission.

Experimental results—We validate the selective excitation profile of the step pulse by creating number states of motion $\ket{n}$, applying the step pulse, and then measuring the internal state excitation probability. The number states are created by first ground state cooling the ion to $\ket{S,0}$ and then using a sequence of BSB $\pi$-pulses of the appropriate duration [46]. The results of the test and the expected excitation profile are shown in Fig. 2b. The overwhelming majority of the errors in the step pulse measurement sequence come from the composite-phase pulse, and not the subsequent state-dependent fluorescence which is performed with error $<10^{-3}$. We mainly attribute deviations from the expected pulse to Rabi frequency drifts caused by vibrations due to the cryostat motor in the experimental assembly. In Fig. 2b, we show simulated results with Rabi frequency noise (estimated from independent measurements) alongside experimental data. We also have contributions from imperfect ground state cooling (which results in lower number state creation fidelity), and smaller contributions from internal state decoherence ($T_{2}\sim 580\mu s$) due to slow B-field fluctuations, and motional state decoherence ($T_{2}\sim 280\mu s$). See the SM for a more detailed error analysis and for details of the setup and control system.

Experimental QFPTD measurement results for $N_{B}=2,3,4$ are shown in Fig. 3. The experimental results—notably their long-time behavior—show good agreement with the theoretical predictions and are consistent with an exponential decay at long times. As $N_{B}$ increases, the ballistic portion is longer, in good agreement with theory. Pulse errors tend to increase the probability of early detection; thereby shifting the QFPTD forward in time. This effect also contributes to an exponential tail with a shorter characteristic time than that of the corresponding theoretical prediction. In order to compare the effect of moving $N_{B}$ without becoming dominated by step pulse errors, in Fig. 3 we scale $\theta$ proportionally to $N_{B}$.

In a separate experiment we explore a critical aspect of stroboscopic FPTDs: the dependence of the escape probability on $\theta$. We compare QFPTDs with $N_{B}=3$ and different $\theta$ in Fig. 4. Since the QFPTDs shown here are probabilities and not probability densities, direct comparison of the QFPTDs for different values of $\theta$ is done using the escape probability. The experimental results are consistent with the presence of an enhancement of the escape probability for smaller $\theta$. However, with current step pulse error rates, these data do not conclusively demonstrate it. This enhancement is a consequence of the repeated measurements within each run. If instead measurements were performed once in a run, then an ‘escape’ probability could be reconstructed, but it would be insensitive to $\theta$.

Conclusions— We develop and validate a novel way to use the motion of a trapped ion to measure QFPTDs. We use this method to measure experimentally the QFPTD of the ion energy when it is coupled to a high-temperature amplitude reservoir, and show good agreement with theory. The step pulse measurements developed here can potentially be used for bosonic state engineering and simulation [62, 63]. Further engineering of bespoke measurement operators raises the possibility of studying the properties of QFPTDs with more exotic surviving domains, such as quantum recurrence times [64] and winding numbers [65, 66]. Complex interaction graphs for applications such as quantum walk search algorithms can be implemented using more co-trapped ions thereby increasing the number of motional and spin degrees of freedom [67, 68, 62]. Using more trapped ions, The role of entanglement in QFPTDs could be investigated, which remains largely unexplored.

We are grateful to Jude Alnas for assistance with ARTIQ control. This work was funded by NSF under QLCI: Center for Robust Quantum Simulation OMA-2120757. This manuscript was edited in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452.