Sensing Out-of-Equilibrium and Quantum Non-Gaussian environments via induced Time-Reversal Symmetry Breaking on the quantum-probe dynamics

The progress on controlling single quantum systems at atomic and nanometric scales has lead to the development of quantum technologies [1, 2, 3]. Both the storage and processing of quantum information in quantum devices suffer from decoherence, the loss of quantum information as a function of time, that distorts the encoded information [4]. Nevertheless, decoherence effects are a key information resource about the environment that is exploited for designing novel quantum sensors with important applications in geology, archaeology, material science, biology and medicine [5, 6, 7] .

These quantum probes have strong potential to enable measurement of physical properties with unprecedented sensitivity, but more importantly they allow probing spatial scales that are not accessible by classical means, such as the atomic and nanoscales [6, 5]. They have already enabled the magnetometry of single neurons [8] and magnetic biomarkers with subcellular resolution [9], microscale and nanoscale detection of single molecules [10, 11], and the probing of temperature-dependent biological processes in cells and small organisms [12].

In the context of these scales, particularly when employing single quantum sensors, environmental systems either manifest intrinsic out-of-equilibrium features or can be unavoidable driven out-of-equilibrium via the quantum feedback induced by the probe, both inducing non-stationary environmental fluctuations [13, 14, 15, 16, 17, 18]. Moreover, quantum probes at these scales can be strongly coupled to its environment generating what is known as non-Gaussian effects [19, 20, 21, 22, 23, 24, 25]. Conventional frameworks to describe the quantum open nature of these sensors do not account for non-stationary environmental features and/or non-Gaussian effects [26, 27, 4, 5]. Only recently, due to progress in quantum sensing technologies, frameworks for sensing out-of-equilibrium and/or non-Gaussian environments with quantum sensors have been introduced [20, 21, 17, 23, 24, 18, 25].

In this article, we delve into the realm of time reversal symmetry to design quantum control sensing-paradigms of the ubiquitous environments found at atomic and nanoscales. We demonstrate the time reversal symmetry breaking in the quantum control of a qubit-sensor, specifically when coupled to quantum non-Gaussian and/or out-of-equilibrium environmental interactions. Leveraging this characteristic, arising from the partial information observed by a quantum-probe, we present a novel quantum sensing paradigm rooted in the design of time-asymmetric dynamical control of the sensor. We name this technique SENSIT (Sensing of Environmental Non-Symmetric Information due to T-symmetry breaking) and demonstrate it through experimental quantum simulations using solid-state NMR.

In this context, we illustrate how the distance of the quantum environmental state from equilibrium can be encoded onto a qubit-probe signal contrast. Furthermore, we showcase the selective filtration of non-stationary features with respect to stationary noise fluctuations. Additionally, we introduce protocols that leverage this contrast to selectively quantify quantum non-Gaussian features and non-equilibrium characteristics of the environment. Overall, our work marks a step forward in the practical application of quantum sensing technology, offering valuable insights into the ubiquitous out-of-equilibrium and quantum non-Gaussian environments encountered by quantum sensors.

To demonstrate the quantum sensing paradigm based on the time reversal symmetry breaking, we consider a dynamically controlled qubit-sensor coupled to an environment that induces pure dephasing. This quantum sensor platform is found in a variety of systems as in electrons in diamonds [28, 29, 11], electronic spins in nanoscale nuclear spin baths [30], quantum dots [31], donors in silicon [32], superconducting qubits [27], trapped atoms [33], and solid-, liquid- and gas-state NMR systems [26, 7].

Control on the qubit-sensor via dynamical decoupling effectively modulates the strength of the qubit-environment interaction and can thus be used to selectively encode environmental information on the signal decay of the qubit-sensor [26, 27]. In the interaction picture with respect to the environmental evolution and control of the qubit, the qubit-environment Hamiltonian is

where $S_{z}$ is the qubit-probe spin operator in the $z$ direction, $B(t)$ is the noise operator representing the environmental fluctuating degrees of freedom that induce dephasing on the qubit-probe, and $f(t)$ is the qubit-environment interaction whose time dependence is only due to the modulation induced by dynamical decoupling [34, 35, 36, 37, 27, 26] (Fig. 1a). The fluctuating noise operator is given by $B(t)=\mathrm{e}^{iH_{E}t}B\mathrm{e}^{-iH_{E}t}$, where $H_{E}$ is the environmental Hamiltonian and $B$ is the environmental degree of freedom coupled to the qubit-probe.

The evolution operator of the qubit-probe is thus $U(T_{s})=\mathcal{T}\mathrm{e}^{-i\int_{0}^{T_{s}}\mathrm{d}t\,f(t)S_{z}B(t)}$, where $\mathcal{T}$ is the time-order superoperator and $T_{s}$ is the sensing time, the time during which the qubit-sensor dephases due to sensing the environmental fluctuations. The probe observable is the in-plane qubit-polarization that decays as $M(T_{s})=\frac{\left\langle S_{+}\otimes\mathbb{I}_{E}\,\rho(T_{s})\right\rangle}{\left\langle S_{+}\otimes\mathbb{I}_{E}\,\rho(0)\right\rangle}=\mathrm{e}^{-\mathcal{J}(T_{s})}$ due to the environment with the decoherence factor $\mathcal{J}$, where $\rho(T_{s})$ is the density matrix of the full qubit-environment system after the qubit has sensed the environment, $\left\langle\cdot\right\rangle=\text{tr}\left[\cdot\right]$, and $S_{+}=S_{x}+iS_{y}$ the up spin operator.

We assume that, in the quantum sensing protocol, the qubit-environment system is initially in a separable state when the probe is brought into contact with the environment. Specifically, we consider the initial state $\rho(0)=\rho_{0}=pS_{x}\otimes\rho_{E}$ with the qubit polarized in the $x$ direction. We do not explicitly write terms proportional to the qubit identity $\mathbb{I}_{S}$ in the density matrix, as they do not contribute to the qubit-probe signal.

We perform a cumulant expansion of the decoherence factor $\mathcal{J}$ on the SE interaction coupling strength with the environment

where $W_{n}$ are the cumulants that completely characterize the environment fluctuations felt by the qubit-sensor. They are defined based on the environmental correlation functions

for $t_{1}\leq t_{2}\leq\dots\leq t_{n}$ with $\left\{\cdot,\cdot\right\}$ the anti-commutator (see SI A).

In quantum mechanics, the evolution operator $U(t)$ is unitary, meaning it is invertible ($U^{\dagger}(t)=U^{-1}(t)$). When a quantum system exhibits time-reversal symmetry, this is typically expressed through the time reversal operator $T$, which is an anti-unitary operator. Time reversal symmetry implies that $T$ commutes with the system Hamiltonian $[T,H]=0$ and thus satisfies $T^{-1}U(t)T=U^{-1}(t)$ [38].

At first glance, one might expect this symmetry to result in a corresponding symmetry on the control operation of the qubit-probe dynamics. However, we here demonstrate that when a dynamically controlled qubit-sensor probes an environment, the partial information accessible to the sensor, obtained through the partial trace of environmental degrees of freedom, can unveil a breaking in the time-reversal symmetry of the control function. In particular, we illustrate that the symmetry is disrupted, when the noise operators of the environment, denoted as $B(t)$, fail to commute at different times. This breakdown occurs explicitly when the decoherence factor involves cumulants of order $n\geq 3$. Moreover, we also show that the symmetry is always broken if the environmental fluctuations are non-stationary, indicative of non-equilibrium features (Proof in SI C).

Quantum non-Gaussian noises thus induce time-reversal symmetry breaking in the quantum control of the qubit-sensor. These noises are characteristic of quantum environments strongly coupled to the sensor and operating at low temperatures. The manifestation of symmetry breaking is absent when the environment is weakly coupled validating the Gaussian approximation, or at the high-temperature limit leading to the semiclassical field approximation (see SI C.2). Furthermore, in the latter scenario, if the environmental fluctuations are non-Gaussian but can be effectively described by classical fields, the time-reversal symmetry remains protected.

A nonstationary environment does not have cumulants invariant under time translation $\Delta t$, i.e. $W_{n}(t_{1},\dots,t_{n})\neq W_{n}(t_{1}+\Delta t,\dots,t_{n}+\Delta t)$. In this case, for quantum or classical noise operators, the time-reversal symmetry of the control is broken. Notice that time translation symmetry is achieved when the environment reaches a stationary state, including when it is in equilibrium. This is illustrated in Fig. 1, which compares stationary fluctuations of the field $B(t)$ (panel b) with non-stationary fluctuations induced by quenches (panels c and d). Time-reversal symmetry –in the control– is broken in the latter cases.

This asymmetry in time evolution is reflected in the dephasing of the qubit-probe (Fig. 1e), where the dephasing near the quench decays slower or faster compared to the stationary case, depending on whether the noise fluctuation variance just before the quench is lower or higher than in the stationary regime. This exemplifies how one can exploit the manifestation of the time reversal symmetry breaking of the environment on the qubit-probe dephasing.

The qubit-probe signal has time-reversal symmetry in the control function if the cumulants in Eq. (2) satisfy the symmetry condition $W_{n}(t_{1},\dots,t_{n})=(\pm 1)^{n}W_{n}(T_{s}-t_{1},\dots,T_{s}-t_{n})$. The sign $(\pm 1)^{n}$ depends on the nature of the noise, i.e. when the noise operator $B$ is an electric charge, electric field, gate potential, etc. the sign corresponds to $1$, and in cases where it is a magnetic field, magnetization, electric current, etc. corresponds to $-1$. We thus call these cases of the electric- and magnetic-type, respectively (details and proof in SI C.2). The cumulants exhibit this symmetry if and only if they possess time translation symmetry, and the noise operators commute at different times or their non-commutation is negligible.

Under these conditions, we prove that the qubit-probe signal satisfies time-reversal symmetry in the control function

where $M_{f}$ is the signal measured when using the control function $f$ and $M_{f_{T}}$ is that measured when using as control the time reversal of $f$, i.e. $f_{T}(t)=f(T_{s}-t)$ (proof in SI C.3). This demonstrates that the polarization of the qubit-probe remains invariant, up to conjugation, under time reversal of the control sequence determined by the control function $f$, when the cumulants exhibit time-reversal symmetry. This time-reversal symmetry is a consequence of stationary correlation functions of an environment plus negligible quantum noncommutativity effects guaranteed only by classical fields, weak coupling, high temperature or Gaussianity. When either of these conditions fails, the symmetry is broken, thus enabling the measurement of quantum non-Gaussianity and/or non-stationary phenomena due to out-of-equilibrium dynamics.

Therefore the argument $\Delta\mathcal{J}=\mathcal{J}_{f}-\mathcal{J}_{f_{T}}^{(*)}$ of the ratio between the corresponding qubit-probe signals $M_{f}/M_{f_{T}}^{(*)}=\exp\left\{-\Delta\mathcal{J}\right\}$ is proportional to the degree of environmental time-reversal symmetry breaking and/or the breaking of the time translation symmetry of the environmental correlation functions (see SI B and C). Here ^(∗) corresponds to the complex conjugation applied only for magnetic-type cases. The ratio $M_{f}/M_{f_{T}}^{(*)}$ is in general a complex number, where its modulus is given by the even cumulants and the phase by the odd cumulants (see SI C.3). Thus the modulus defines the SENSIT qubit-signal contrast, and the phase, the SENSIT qubit-phase contrast (see SI B). In this article, our emphasis is on the SENSIT qubit-signal contrast due to its greater accessibility and robustness in experiments. However, in principle, either quantity can be employed.

The SENSIT qubit-signal contrast is

where

is the difference between the $2n-$th cumulant at forward times $t_{i}$ and at the time reversed ones from the sensing time $T_{s}-t_{i}$. These quantities serve as order parameters, gauging the extent of time-reversal symmetry breaking in the qubit-control. The SENSIT contrast, therefore, acts as a probe for these order parameters, with weights determined by the control function $f(t)$. Notice that in Eq. (5), we performed a change in the time variables to convert the reversed control $f_{T}(t)$ into $f(t)$, and this change the arrow of time in the cumulants $W_{2n}(T_{s}-t_{1},\dots,T_{s}-t_{2n})$.

The key to utilizing this quantification lies in its resilience against any noise contribution to qubit-probe dephasing that does not induce a time-reversal symmetry breaking, i.e. a contribution that is stationary and such that the noise operators either commute at different times or their non-commutation is negligible. This robustness is ensured as the terms from $W_{2n}(t_{1},\dots,t_{2n})$ cancel out with those in $W_{2n}(T_{s}-t_{1},\dots,T_{s}-t_{2n})$ and similarly for the phase term (proof in SI C.3). This means that if the sensed environment can be separated into two independent parts, $a$ and $b$, such that only $a$ induces a time-reversal symmetry breaking on the control, then the SENSIT contrast will exclusively sense properties of $a$; while remaining completely independent of $b$. Consequently, the SENSIT contrast exclusively responds to noise sources that induce the time-reversal symmetry breaking of the qubit-probe control (SI D). This may set an avenue for pump and probe experiments, where a qubit that interacts with a complex environment is used to sense just a part of it by first driving the desired target subsystem out of equilibrium (pump) and then use SENSIT to selectively detect just that subsystem (probe).

To give an example, the SENSIT contrast provides information about the distance to equilibrium. In the simple but general case when the environment state $\rho_{E}=\rho_{E}^{(0)}+\epsilon\rho_{E}^{(1)}$ is near to a stationary state $\rho_{E}^{(0)}$, with $\rho_{E}^{(1)}$ a constant perturbation, we demonstrate that $\Delta\mathcal{J\propto\epsilon}$ thus quantifying the distance to equilibrium (see SI E). In the particular cases illustrated in Fig. 1, the SENSIT contrast $\text{Re}\Delta\mathcal{J}\propto\left(\sigma-\sigma_{0}\right)$, where $\sigma$ the variance of the noise fluctuation before the quench, and $\sigma_{0}$ the variance at the stationary regime achieved at long times after the quench (see SI F). This thus sets a paradigmatic example about how the SENSIT contrast is proportional to the distance $\left(\sigma-\sigma_{0}\right)$ of the initial state of the environment to its stationary state at equilibrium.

One of our main results is that the SENSIT contrast, being a readily available quantity in experimental setups, serves as a sensing protocol for detecting and characterizing out-of-equilibrium and/or quantum non-Gaussian environments. To provide a concrete illustration of the underlying principles of SENSIT and enhance the clarity of the introduced general results, we experimentally performed solid-state NMR quantum simulations on a Bruker Avance III HD 9.4T WB NMR spectrometer with a ${}^{1}\text{H}$ resonance frequency of $400.15$ MHz and a ${}^{13}\text{C}$ resonance frequency of $100.61$ MHz. The ${}^{13}C$ nucleus plays the role of the qubit-probe, and the surrounding ${}^{1}H$ nuclei are considered the environment. Since ${}^{13}C$ is in natural abundance, it is present in low concentration and all interaction between ${}^{13}C$ nuclei are negligible on the performed experiments (see SI G.1 for details on the experimental setup).

The system is initially in thermal equilibrium, as represented in the first step of the sensing protocol in Fig. 2a. We induce an out-of-equilibrium state in the environment to generate non-stationary noise fluctuations on the qubit-sensor. To achieve this, we initially employ the qubit-probe to build up quantum correlations between the qubit and the environmental spins during a preparation time $T_{p}$, ensuring a localized spread of information near the sensor (Fig. 2a,b).

Then we performed a quantum nondemolition measurement on the qubit-probe state mimicked by induced dephasing to erase the probe-environment correlations [39, 7]. This procedure leaves the environment in a correlated out-of-equilibrium state (see third step of Fig. 2a and SI G.2). All these steps constitute the preparation of a nonequilibrium state in the environment.

Subsequently, we initialize the qubit-probe state in a separable state with the environment $\rho_{0}=pS_{x}\otimes\rho_{E}$ by the application of a $\pi/2$ pulse on the qubit. After the initialization, we proceed to implement non-equidistant dynamical decoupling sequences to manipulate the time-reversal symmetry of the sequence. We observe the decay of the qubit signal at the sensing time $T_{s}$, representing the duration of the dynamical decoupling sequence, to quantify the SENSIT contrast. This process is illustrated in the last two steps of Fig. 2a.

We employed the Selective Dynamical Recoupling (SDR) sequence [40, 41], chosen for its simplicity in constructing a time-asymmetric sequence using only $\pi$-pulses, along with the flexibility of having a single parameter that can be adjusted without changing $T_{s}$. This makes SDR the most straightforward choice for our purposes. It consists of a concatenation of a CPMG spin-echo train with $N-1$ rapid, spin-echo train $\pi-$pulses between the times $t=0$ and $t=xT_{s}$, and a Hahn spin-echo sequence consisting of a single echo $\pi-$pulse at the center between the times $xT_{s}$ and $T_{s}$. The SDR modulation $f(t)$ is shown with the blue curve in Fig. 2a. Here $x$ is a dimensionless parameter, that defines the asymmetry of the SDR sequence interpolating between a single Hahn echo at $x=0$ and a CPMG sequence of $N$ equidistant pulses at $x=\frac{N-1}{N}$. The time-reversed SDR (TSDR) sequence contains the inverse succession in time of the $\pi-$pulses, consisting first of the single Hahn-echo sequence between times $0$ and $(1-x)T_{s}$, followed by the $N-1$ CPMG pulses between the times $(1-x)T_{s}$ and $T_{s}$ (see the orange curve for $f_{T}(t)$ in Fig. 2a and SI G.2).

The insets in Fig. 2c illustrate the qubit-probe signal following the SDR and TSDR modulations as a function of $x$, for a fixed total sensing time and varying preparation times $T_{p}$ for the initial out-of-equilibrium state in the environment. The results demonstrate that the SDR and TSDR signals are indistinguishable at $T_{p}=0$ when the environment is stationary at equilibrium. Subsequently, they showcase how the signals progressively increase their contrast as the environment is shifted further out of equilibrium producing non-stationary noise fluctuations.

To get a single SENSIT-contrast quantification of the non-equilibrium degree of the environmental state, we integrate the attenuation factors of the qubit signals $\int_{0}^{\frac{N-1}{N}}\mathrm{d}x\,\text{Re}\Delta\mathcal{J}(x)$ over the parameter $x$ (main panel of Fig. 2c). This contrast is proportional to the distance of the initial environmental state from equilibrium, consequently increasing as a function of the preparation time, with a greater number of environmental spins becoming correlated (see SI C).

These experiments were performed at room temperature that represents a high-temperature limit as the thermal energy is much larger than the Zeeman energy of the spins (see SI G.1), therefore the time-reversal symmetry for quantum control described in Eq. (4) will only be broken if the environment is out-of-equilibrium manifesting non-stationary noise fluctuations. Thus here we demonstrate how the SENSIT contrast probes a quenched state on the environment and determines how far from equilibrium it is [13, 24, 25].

To have an alternative method for quantifying the out-of-equilibrium degree, we employed a more established approach based on out-of-time-order correlations (OTOC) measured with multiple quantum coherences [15, 42]. This method assesses the non-commutation degree of the evolved density matrix during the preparation time in relation to the initial state of the environment before preparation, as depicted in Fig. 2b. This OTOC approach quantifies the effective number $K$ of environmental spins that were correlated during the preparation step [13, 43] (see SI G.3 for details).

Figure 2c compares the number $K$ of correlated environmental spins to the qubit-probe during preparation, and the SENSIT contrast measured from the qubit-probe. Both quantities increase with the preparation time, manifesting a monotonous relation between the established OTOC measure $K$ of non-equilibrium degree and the proposed SENSIT contrast. Measuring the OTOC involves experimental control over the environment, necessitating the ability to apply collective rotations to environmental spins and reverse the many-body evolution resulting from environmental spin-spin interaction (see SI G.3). Consequently, while OTOC measurements are feasible in e.g. NMR experiments, they are not readily available for most systems. In contrast, assessing the SENSIT contrast only requires control over the qubit-probe, specifically without the need for any control over the sensed environment.

To illustrate how the SENSIT contrast, reliant on time-reversal symmetry breaking, can selectively captures information about out-of-equilibrium or quantum non-Gaussian environmental fluctuations over stationary noise fluctuations, we measure the effect of environmental scrambling [15, 42]. In this context, we examine how the quenched state information is erased due to scrambling, consequently impacting the SENSIT contrast (Fig. 3).

Quantum information scrambling is the encoding of an initial local information into non-local degrees of freedom, in this case due to the environmental dynamics driven by the Hamiltonian of the environment $H_{E}$ (Fig. 3b) [15, 42]. This in turn renders the information inaccessible by local measurements, and it is related to the autothermalization and quantum information dynamics of the environment [13, 44, 45, 16, 15, 46, 43, 42]. Since the qubit-probe performs a local measurement, the measurements after scrambling should match those found at equilibrium. The key control source in this context involves introducing a waiting period, denoted as the environmental scrambling time $T_{E}$, before implementing the detection of the SENSIT contrast following the quenching of the environment (Fig. 3a). In the case of a non-Gaussian environment that is stationary, the cumulant expansion terms exhibit time translation symmetry, rendering the SENSIT contrast invariant with respect to the waiting time. Conversely, an out-of-equilibrium environment relaxes towards equilibrium as a function of the waiting—scrambling—time.

The decay of the SENSIT contrast, observed in Fig. 3c, is presented as a function of the environmental scrambling time $T_{E}$. The insets provide the SDR and TSDR signals for three distinct scrambling times. The contrast between the SDR and TSDR curves diminishes as the quenched state is progressively scrambled away. These results thus effectively showcase SENSIT’s ability to selectively quantify the extent to which the environment of the qubit-probe is departing from equilibrium and evolving towards equilibrium. Importantly, SENSIT enables the measurement of quantum information scrambling without relying on environmental time reversions, a requirement typical need in measurements based on OTOCs [15, 42].

While most quantum systems exhibit time reversal symmetry [38], our research unveils a fundamental breakdown of time reversal symmetry when a quantum sensor probes partial information from environments, characterized by out-of-equilibrium, non-stationary dynamics or with interactions containing quantum non-Gaussian correlations. This opens a quantum sensing paradigm, offering a lens to explore ubiquitous environments at the quantum level, where intrinsic out-of-equilibrium dynamics prevail, driven either by inherent fluctuations or quantum feedback induced by the probe [17, 24, 18, 25]. Moreover, it offers a tool for single sensors at atomic or nanoscales that can be strongly coupled with the environment, thus generating quantum non-Gaussian effects on the dephasing of the qubit-probe [25].

Our findings gain practical significance in the realm of quantum sensing. Specifically, within pump and probe schemes [47], our results open avenues for tailored measurements using qubit-probes, enabling selective detection of the environmental degrees of freedom being selectively pumped out of equilibrium. Moreover, our results holds promise for characterizing noise of time crystals [48] or Floquet systems [49], specially those with partially broken time translation symmetry induced by external driving forces. This offers insights into the dynamics of such systems, providing a means to understand and quantify the intricate interplay between the quantum probe and environments with complex, time-varying characteristics [14, 24, 25].

An additional strength of our work lies in its ability to capture information scrambling of local operators, like the OTOCS, but notably without requiring environmental time reversions [15, 42]. This unique feature positions our research as a valuable tool for detecting persistent states characterized by long-lasting temporal features, such as localized modes [50]. In essence, our study not only contributes theoretical insights but also introduces practical methodologies that can be harnessed to explore and manipulate quantum systems in diverse and dynamic environments.

In conclusion, our study not only contributes fundamental insights into the breakdown of control function time reversal symmetry in quantum sensing but also presents a platform for practical applications. From tailored measurements in complex samples to probing intriguing phenomena in time crystal and Floquet systems, the potential impact of our findings offers an alternative avenue in the realm of quantum sensing technologies. This work paves the way for future research directions and applications, emphasizing the dynamic interplay between quantum sensors and complex, out-of-equilibrium environments.

This work was supported by CNEA; CONICET; ANPCyT-FONCyT PICT-2017-3156, PICT-2017-3699, PICT-2018-4333, PICT-2021-GRF-TI-00134, PICT-2021-I-A-00070; PIP-CONICET (11220170100486CO); UNCUYO SIIP Tipo I 2019-C028, 2022-C002, 2022-C030; Instituto Balseiro; Collaboration programs between the MINCyT (Argentina) and, MAECI (Italy) and MOST (Israel).