Intrinsic and induced quantum quenches for enhancing qubit-based quantum noise spectroscopy

Quantum sensing protocols that exploit the dephasing of a probe qubit are powerful and ubiquitous methods for interrogating an unknown environment. They have a variety of applications, ranging from noise mitigation in quantum processors, to the study of correlated electron states. Here, we discuss a simple strategy for enhancing these methods, based on the fact that they often give rise to an inadvertent quench of the probed system: there is an effective sudden change in the environmental Hamiltonian at the start of the sensing protocol. These quenches are extremely sensitive to the initial environmental state, and lead to observable changes in the sensor qubit evolution. We show how these new features give access to environmental response properties. This enables methods for direct measurement of bath temperature, and for detecting non-thermal equilibrium states. We also discuss how to deliberately control and modulate this quench physics, which enables reconstruction of the bath spectral function. Extensions to non-Gaussian quantum baths are also discussed, as is the application of our ideas to a range of sensing platforms (e.g., nitrogen-vacancy (NV) centers in diamond, semiconductor quantum dots, and superconducting circuits).

In the main text, we have focused on the specific type of quantum noise spectroscopy measurements where a sensor qubit is coupled to its environment via pure-dephasing-type interactions. Within this setting, we have shown the quench phase shift (QPS) lets one probe the response properties, or spectral function of the environment, which would be otherwise inaccessible using standard dephasing-based noise spectroscopy.
Interestingly, in principle one can extract similar information about the imaginary part of response function ImG R ξξ [ω], or equivalently the spectral function, using an extended version of standard T 1 relaxometry experiments. Conventional T 1 -type experiments specifically probes transversely coupled bath fields (e.g., viaĤ int =σ x ⊗ξ), which induces transitions between the qubit levels. In this setting, typically one would measure the qubit population decay rate Γ tot ≡ 1/T 1 [1], which corresponds to the sum of qubit relaxation and excitation rates. A straightforward calculation based on Fermi's Golden rule can then relate Γ tot to the symmetrized noise spectral density (NSD) via Γ tot = 2S[Ω], whereas the difference between relaxation and excitation rates corresponds to the response function ImG R ξξ [Ω] (see the following paragraph for more detail). While the bath NSD can be directly inferred from T 1 -decay rate Γ tot , to further probe response function ImG R ξξ [ω] one would also need to measure the qubit steady state population σ z ss : the latter measurement is not a part of standard T 1 relaxometry [2,3]. In comparison, the QPS in Eq. (17) is readily accessible in standard T 2 -type measurements, and as we show in the main text, arguably offers a more direct knob to probe the response properties of longitudinal bath fields. We also note that standard T 1 -type experiments are not sensitive to dephasing baths. Although in principle one can use the spin-locking technique (also known as T 1ρ measurements) [2], i.e. continuously drive the qubit to measure longitudinal bath fields via relaxometrybased experiments, in practice the drive strength needs to be higher than the inhomogeneous linewidth of the qubit. Thus, the range of frequencies that can be probed using the spin-locking technique often does not correspond to the dominating dephasing source for the undriven qubit (see e.g. [4][5][6]). In contrast, T 2 -type experiments with QPS measurements are more suitable for probing low-frequency dephasing noise source.
We now concretely show the relation between qubit decay rate Γ tot and the steady state population σ z ss to environmental properties. For the case of transverse coupling to the bath, it is more illuminating to represent bath properties in terms of the quantum noise spectra S[ω] ≡ +∞ −∞ dte iωt ξ (t)ξ(0) , so that the Fermi's Golden rule transition rates for qubit excitation and relaxation Γ ± are given by Γ ± = S[∓Ω] (Ω denotes qubit transition frequency; see Ref. [7] for a pedagogical introduction). One can use a few lines of algebra to show that the symmetrized and anti-symmetrized components of S In the main text, we present the asymptotic long-time behavior of qubit dephasing function and quench phase shift (QPS) in Eqs. (28) and (29), assuming that the bath noise spectral density (NSD) and density of states functions exhibit power-law dependence in the asymptotic low-frequency limit. For clarity, in this Supplementary Note, we provide a detailed derivation of the asymptotic results. We start with the general expressions for the dephasing function ζ(t f ) and QPS Φ q (t f ) under any spin-echo qubit control pulse, given by Eqs. (16) and (20) in the main text as We also assume that the bath NSDS[ω] and response functions ImG R ξV [ω] exhibit power-law dependence in the asymptotic low-frequency regime (see Eqs. (25) and (26) in the main text) For convenience, we rewrite the bath NSDS[ω] and response functions ImG R ξV [ω] in the full frequency range in terms of cutoff functions µ A (x) (A = S, G) asS where we introduce a UV cutoff frequency ω c below which the asymptotic power-law function provides a good approximation for the exact function. By definition, the cutoff functions µ A (x) satisfy following conditions and we further assume both cutoff functions µ A (x) (A = S, G) corresponding to any physical bath are smooth near x = 0. We now consider a generic qubit control pulse satisfying F [0] = 0, which consists of L instantaneous π-pulses at times t = α t f . Without loss of generality, we assume the coefficients α ( = 1, 2, . . . , L) satisfy following conditions so that we can explicitly compute the filter function F [ω] as Substituting above equation into Supplementary Eqs. (1) and noting that the integrands are even functions of frequency, we obtain In the long-time limit t f → +∞, the integrals above would tend asymptotically to universal limits that are independent of details about the physical cutoffs, if and only if the integrals when setting µ A (x) ≡ 1 (A = S, G) are well defined. For this scenario, the asymptotic limits of dephasing function and phase shift functions can be derived as where the dimensionless coefficients C ζ (p) and C Φ (s) are determined by the spin-echo pulse parameters as and Γ(·) is the gamma function. For Hahn echo, the control pulse parameters are L = 1, α 1 = 1 2 , and substituting the parameters into equations above lets us obtain the coefficients C ζ,H = 1−2 p+1 π Γ (p − 1) sin pπ 2 and C Φ,H = 1−2 s π Γ (s − 1) cos sπ 2 in the main text. Note that above equations are still well-defined if p, s are exact integers, where the gamma function in Supplementary Eqs. (9) alone might diverge: in this case, we could obtain the asymptotic coefficients by taking the continuous limit of Supplementary Eqs. (9) as the exponent approaches the corresponding integer value. The asymptotic limit of quench phase shift can be further simplified if the response function exponent take the value of 1, as For exponents beyond the range of validity specified in Supplementary Eqs. (8), the long-time behavior of the dephasing function (phase shift) may not have a well-defined asymptotic limit, or the asymptotic behavior would depend on details of the low-or high-frequency cutoff of the bath NSD (response function). To illustrate this, we discuss a concrete example where the long-time phase shift dynamics explicitly depends on details of the cutoff. We compare the Hahn echo phase shift dynamics for response function ImG R ξV [ω] = −(A 0 /2)ω s µ G (ω/ω c ) with exponent s = 5/2, and two different UV cutoff functions: exponential cutoff with µ G,exp (x) = e −x , and step-function cutoff with µ G,sp (x) = Θ(1 − x), where Θ(·) is the Heaviside step function. The quench phase shift is generally given by Supplementary Eq. (7b), which for Hahn echo can be computed analytically to yield While the asymptotic t f ω −1 c limit of Hahn echo phase shift Φ q,exp (t f ) assuming exponential cutoff agrees with the universal result in Supplementary Eq. (8b), it is straightforward to see that the phase shift dynamics with step-function cutoff does not have a well-defined asymptotic long-time limit, and does not agree with Supplementary Eq. (8b). Although the oscillatory behavior of the first term in the square bracket in Supplementary Eq. (14) is typical when we have response functions with a step-function cutoff (e.g. see the dashed blue curve in Fig. 5c in the main text, depicting the QPS for Ohmic bath spectral function with a step-function UV cutoff), for exponents within the range of validity of Supplementary Eq. (14) such oscillations are negligible in the asymptotic long-time limit. However, as shown in Supplementary Eq. (14), for exponents outside this range the oscillatory contribution is important even in the long-time limit. Generally, for bath NSD (response function) of the form given by Supplementary Eqs. (2) with exponent p ≥ 1 (s ≥ 2), the long-time behavior of the dephasing function ζ(t f ) (quench phase shift Φ q (t f )) depend on the detail of the UV cutoff of the spectrum. Similarly, for exponents p ≤ −3 (s ≤ −2) below the regime of validity in Supplementary Eq. (8), the corresponding long-time behavior would depend on the low-frequency cutoff.

Supplementary Note 3. Alternative derivation of quench phase shift in Ohmic environments
As discussed in the main text, specifically for baths that exhibit Ohmic behavior (a flat NSD and a linear bath spectral function) in the asymptotic low-frequency limit, i.e., satisfying Supplementary Eq. (2) with p = 0, s = 1, the quench phase shift (QPS) under spin-echo or dynamical-decoupling control sequences tends to a constant in the longtime regime, as shown in Eq. (31) (see also Supplementary Eq. (10) in Supplementary Note 2). In this Supplementary Note, we provide an intuitive derivation of this result, which for a generic control sequence can be written as where We start with the general linear response formula for QPS, assuming quench operator V =ξ/2, quench function η(t) = Θ(t)Θ(t f − t), and a generic filter function, which in the time domain is given by (see Eq. (77) in the main text) We can rewrite the expression on the right hand side using integration by parts as The first term on the RHS can be viewed as the net phase shift due to a constant qubit frequency shift t f 0 dtG R ξξ (t) accumulated during the time evolution, whereas the second term accounts for a residual phase correction due to the fact that the quench-induced frequency shift to the qubit is time dependent. For Ohmic baths whose response functions exhibit linear dependence in the asymptotic low-frequency regime, it is straightforward to show that the asymptotic long-time behaviors of these two terms are given by Thus, the asymptotic long-time behavior of QPS with Ohmic baths can be viewed as the sum of phase shift due to a static frequency shift in the long-time limit, which is proportional to F [0], and a residual phase correction. Noting that the bath spectral function J [ω] is related to the response function via J Specifically for dynamical-decoupling-type control pulses with F [0] = 0, the first term would vanish, and we recover Eq. (31) in the main text. As a result, for approximately Ohmic baths with spectral function satisfying J [ω] ∼ ω at low frequencies, the asymptotic behavior of QPS under spin-echo control pulses in the long time t f → ∞ regime is universal (i.e., it only depends on the asymptotic linear dependence of the spectral function), and is independent of the specific UV cutoff of the response function and details of the qubit control sequence.
Supplementary Note 4. Discussion on the use of Hahn echo versus Ramsey coherence times in quench-enhanced QNS for Ohmic bath thermometry In the main text, we show that our quench-enhanced T 2 -style quantum noise spectroscopy offers a direct route (i.e. without any curve fitting) to estimating the temperature of any baths exhibiting Ohmic behavior in the asymptotic low-frequency limit, i.e.S[ω] ∼ const. and J [ω] ∼ ω as ω → 0 + . However, realistic systems may also experience a large amount of quasistatic noise, leading to deviations from perfect Ohmic behavior at infinitesimal frequencies. In this Supplementary Note, we discuss how our thermometry protocol also works in the presence of such quasistatic noise.
Recall that our protocol can be summarized in Eq. (32) in the main text, where we can extract bath temperature T from the quench phase shift Φ q (t f ) and low-frequency noise spectral densityS[0], via the following relation Note that this result assumes the NSD is flat in the low-frequency limit, in which case the qubit Ramsey and Hahn-echo coherence times are necessarily identical. Here we stress that even in the circumstances where the qubit Hahn-echo time T 2 differs from the Ramsey coherence time T FID , a modified version of Supplementary Eq. (21) is still applicable, as long as the slow noise disrupting Ohmic NSD behavior emerges at a much lower frequency scale compared to the Ohmic regime. More specifically, this means the NSDS[ω] and the spectral function J [ω] has a low-frequency cutoff ω ir , below which the Ohmic behaviorS[ω] ∼ const. and J [ω] ∼ ω breaks down. As mentioned, this includes the common physical situations, where the environment also has a large amount of quasistatic noise, which can be described as an additional delta function peak in the NSD. It then follows that, our thermometry protocol is applicable to baths with asymptotic low-frequency Ohmic behavior, which may exhibit a high-as well as a low-frequency cutoff. For this more general scenario, we should use asymptotic low-frequency NSD lim ω→0S [ω] = 2/T 2 , instead of strictly zero-frequency noiseS[0] = 2/T FID in Supplementary  Eq. (21). This justifies the use of Hahn-echo coherence time T 2 in the main text.
Supplementary Note 5. Case study: Quench phase shift generated by electromagnetic environment due to a driven damped cavity mode In the main text, we have considered a quantum bath whose spectral function is asymptotically Ohmic in the lowfrequency limit (i.e.S[ω] ∼ const. and J [ω] ∼ ω as ω → 0 + ) and also exhibits a Lorentzian peak at a finite frequency; the Hahn-echo quench phase shift (QPS) dynamics due to this bath is illustrated in Fig. 5b. As mentioned in the main text, such spectral function can describe dephasing environments generated by a driven damped electromagnetic (EM) cavity. In this Supplementary Note, we provide a detailed discussion on the corresponding physical system.
Consider a qubit dispersively coupled to a driven damped bosonic mode b (resonance frequency ω c , decay rate κ) via the HamiltonianĤ For instance, we may have a superconducting transmon qubit coupled to a microwave cavity mode; the photon shot noise fluctuations due to the cavity mode then induce qubit frequency shift and dephasing during time evolution.
Transforming to the interaction picture defined by the free qubit Hamiltonian Ωσ z /2, as well as frame rotating at the drive frequency ω dr of the cavity mode, the dynamics of the total system can be described by the quantum master equation as followsρ wheren th is the thermal photon number. For simplicity, we assume zero temperature (n th = 0) hereafter, but we stress that our approach also applies to the case with finiten th . In the above equation,Ĥ 0 andĤ dr are rotating-frame Hamiltonians accounting for free cavity dynamics and the linear cavity drive, respectively, aŝ where ∆ ≡ ω dr − ω c denotes the cavity detuning, and f dr is the drive strength. We introduce a dimensionless envelope function y(t) to encode possible time dependence of the drive. For the purpose of our discussion, we can assume the cavity drive is switched on at some earlier time before the start of the Hahn-echo protocol, so that we have y(t) = 1 during the protocol (0 < t < t f ). For convenience, we define the stationary intracavity driven photon number in the absence of the qubit (i.e., setting λ = 0 in Supplementary Eq. (23)) asn dr , so that we have (here · denotes stationary state expectation values of bath operator) (25) In this specific setup, the photon shot noise coupled to the qubit is generally non-Gaussian [11]. Thus, in order to apply our results in the main text (see also Supplementary Eqs. (1)) to describe the photonic environment, we first need to ensure non-Gaussian effects are small. Without loss of generality, we focus on the mean-field regime where the Gaussian approximation is well justified, i.e. we have approximately which holds if we require parameters to satisfy following conditions n dr > 1, √n * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Supplementary Fig. 1: Numerically simulated Hahn-echo qubit phase shift and the quench phase shift dynamics, corresponding to the photonic environment generated by a driven, damped cavity mode. The purple triangles (green circles) depict qubit phase shifts Φ ↓ (t f ) (Φ ↑ (t f )) using initial qubit state |↓ (|↑ ), which are computed numerically from solving the qubit-cavity master equation in Supplementary Eq. (23). The difference between the two qubit phase shifts [Φ ↓ (t f ) − Φ ↑ (t f )] (dark blue asterisks) in the long-time limit can be well described by the asymptotic expression based on quench phase shift in Supplementary Eq. (35), as expected. Parameters: ∆/κ = 5, λ/κ = 0.5,n dr = 10, andn th = 0.
Protocol time t f (2πκ -1 ) 〈σ Supplementary Fig. 2: Numerically simulated Hahn-echo qubit coherence function σy(t f ) corresponding to the photonic environment generated by a driven damped cavity mode. The purple triangles (green circles) depict qubit Hahn-echo coherence function σy(t f ) using initial qubit state |↓ (|↑ ), which are computed numerically from solving the qubit-cavity master equation in Supplementary Eq. (23). Note that data points with small protocol times (t f < π/κ) fall out of range of the figure and are not shown here; however, the short-protocol-time behavior is determined by the high-frequency components of the spectral function, and does not affect our conclusion. The parameters are the same as in Supplementary Fig. 1.
and making use of solution to the master equation in Supplementary Eq. (23), we can straightforwardly obtain the NSD asS [ω] =n dr λ 2 Similarly we can derive the bath spectral function J [ω] as Note that we can rewrite the spectral function in Supplementary Eq. (30) as which recovers the form of spectral function used to compute data shown in Fig. 5b, if we redefine centers of Lorentzian peaks as ±∆. It is straigtforward to check that the photon shot noise exhibits Ohmic behavior in the asymptotic low-frequency limit, i.e. we haveS [ω] ∼n dr λ 2 κ κ 2 Making use of results in Supplementary Eqs. (8), we thus obtain the asymptotic behavior of dephasing function ζ(t f ) and the quench phase shift Φ q (t f ) We now compare the predicted asymptotic results to exact dynamics from directly solving the master equation in Supplementary Eq. (23). We consider qubit dynamics corresponding to standard Hahn-echo protocol, where the qubit-bath system is initialized as follows: i) we first prepare the qubit in one of eigenstates, |↑ or |↓ ; ii) we then switch on cavity drive, and wait for long enough so that the cavity reaches a stationary state at the start (t = 0) of the Hahn echo protocol. Denoting the qubit phase shift corresponding to initial state |↑ (|↓ ) as Φ Because the master equation conserves qubit polarization (i.e.,σ z ), and is quadratic in terms of bosonic mode operators, we can numerically simulate the exact qubit-cavity system dynamics efficiently (see [11] for details). Supplementary Fig. 1 illustrates the exact qubit phase shift dynamics corresponding to initial qubit state |↓ and |↑ (purple triangles and green circles, respectively) for the choice of parameters ∆/κ = 5, λ/κ = 0.5, andn dr = 10. As shown in Supplementary Fig. 1 indicates that low-frequency photon shot noise has a finite temperature. At first glance, this might seem surprising, as the system master equation in Supplementary Eq. (23) only includes a purely cooling dissipator when we assume zero thermal photon number (n th = 0), and does not involve any explicit heating. We note that there is in fact no contradiction: the finite temperature of low-frequency fluctuations reflects the fact that Markovian dissipation actually corresponds to a non-equilibrium environment, and the quench approach provides a direct knob to probe this physics.
Supplementary Note 6. General strategy for reconstructing the environmental spectral function in a generic frequency range using time-dependent quench functions In the main text, we discussed using sensor qubits based on a single nitrogen vacancy center in diamond to engineer a time-dependent quench (c.f. Fig. 6), and we discussed its application in reconstructing the bath spectral function for a specific type of control pulses. In this Supplementary Note, we discuss a general recipe to construct more general periodic control pulses, which lead to a powerful set of varying spectral filters that can be utilized to reconstruct the spectral function J [ω] in a broad range of frequencies.
As discussed in the main text (see discussions following Eq. (33)), to illustrate the idea we focus on case where the quench operator is directly related to noise, withV =ξ/2. Without loss of generality, we also focus on periodic NV control pulses, which are suitable for reconstructing the spectral function at finite target frequencies. Recall that the spin-1 structure of the NV lets us effectively realize a nontrivial quench function η(t), in addition to the standard noise filter function. More specifically, we can apply a periodic sequence of NV control pulses (period T with 2M repetitions, M ∈ Z), switching between the qubit subspaces {m z = 0, m z = +1} and {m z = 0, m z = −1} (see also Fig. 6 in the main text), to realize a periodic quench function as where η 0 (t; 2T ) denotes the base quench function, and satisfies the following relation The structure of switching pulses also ensure that η 0 (t; 2T ) = (−) N η 0 (t + T ; 2T ) for 0 < t < T , where N is the total number of switching pulses per period T . For the example control pulse sequence depicted in Fig. 6, we have N = 1 and η 0 (t; where Θ(·) denotes the Heaviside step function. Again introducing the total protocol time satisfying t f = 2M T , we can straightforwardly rewrite Fourier transform of the quench function as For reasons that will become clear, we also assume a periodic sequence of standard qubit control π-pulses, with a same period T and total evolution time t f = 2M T , so that we similarly have We are now ready to present the recipe, or the necessary and sufficient conditions, to construct spectral filters that specifically probe the bath spectral function J [ω] (see also Eq. (22) in the main text) We essentially require that the base filter and quench functions exhibit the same periodicity, and satisfy the following conditions • The base filter and quench functions must be mirror symmetric or anti-symmetric with respect to t = T , i.e., F 0 (t; 2T ) = s F F 0 (2T − t; 2T ), and η 0 (t; 2T ) = s η η 0 (2T − t; 2T ), where s F , s η = ±1.
• The base filter and quench functions exhibit opposite mirror symmetries with respect to t = T , i.e., s F = −s η = +1 or −1. The spectral filter F J [ω; t f ] for J [ω] forms a comb-like structure in frequency space, if we fix pulse periodicity T = t f /2M and take the asymptotic large pulse number limit, i.e.
Thus, given a finite target frequency range, we can construct a corresponding set of NV control pulses that specifically realize frequency comb filters for the spectral function at target frequencies. We can then measure the quench phase shifts under these control pulses in the comb limit (fix T = t f /2M and choose M 1), which in turn enable reconstruction of the spectral function J [ω] via Supplementary Eq. (44).