Enhanced precision bound of low-temperature quantum thermometry via dynamical control

High-precision low-temperature thermometry is a challenge for experimental quantum physics and quantum sensing. Here we consider a thermometer modeled by a dynamically-controlled multilevel quantum probe in contact with a bath. Dynamical control in the form of periodic modulation of the energy-level spacings of the quantum probe can dramatically increase the maximum accuracy bound of low-temperatures estimation, by maximizing the relevant quantum Fisher information. As opposed to the diverging relative error bound at low temperatures in conventional quantum thermometry, periodic modulation of the probe allows for low-temperature thermometry with temperature-independent relative error bound. The proposed approach may find diverse applications related to precise probing of the temperature of many-body quantum systems in condensed matter and ultracold gases, as well as in different branches of quantum metrology beyond thermometry, for example in precise probing of different Hamiltonian parameters in many-body quantum critical systems. Progress in quantum technologies calls for increased precision in probing quantum systems. The authors present a method for substantially improving the precision bound of ultra low-temperature thermometry via dynamical control of a quantum thermometer.

P recise probing of quantum systems is one of the keys to progress in diverse quantum technologies, including quantum metrology 1-3 , quantum information processing (QIP) 4 , and quantum many-body manipulations 5 . The maximum amount of information obtained on a parameter of a quantum system is quantified by the quantum Fisher information (QFI), which depends on the extent to which the state of the system changes for an infinitesimal change in the estimated parameter [6][7][8][9][10] . Ways to increase the QFI, thereby increasing the precision bound of parameter estimation, are therefore recognized to be of immense importance 11,12 . Recent works have studied QFI for demonstrating the criticality of environmental (bath) information 13 , QFI enhancement in the presence of strong coupling 14 or by dynamically controlled quantum probes 9 , and the application of quantum thermal machines to quantum thermometry 15 .

Results
Model. An example of our dynamically controlled quantum thermometer (DCQT) is a quantum wavepacket trapped in a potential and subjected to periodic modulation, while it is immersed in a thermal bath (Fig. 1a). Measurements of the wavepacket after it has reached a steady-state provide information about the bath temperature. Another example of a DCQT is the internal state of either a two-level or a multilevel system coupled to a spin-chain bath 32,33 . The level separation is periodically modulated by a control field (e.g., a field-induced alternating current Stark shift) and its level populations are finally read out by laser-induced fluorescence in the optical range (Fig. 1b). Many of the advanced thermometers [14][15][16][28][29][30][31] may be adapted to such dynamically controlled operation.
The proposed DCQT is described by a Hamiltonian HðtÞ, subjected to a modulation with time period τ ¼ 2π=Δ: Hðt þ τÞ ¼ĤðtÞ ¼ X kĤ k ðtÞ; The DCQT is coupled to a bath through the interaction Hamiltonian of the formĤ I ¼ P kĤIk ¼ P kŜk B k . Here,Ŝ k andB k are, respectively, the k-th mode DCQT and bath operators. In cases where the k-th bath mode acts as a Markovian environment characterized by a mode-dependent unknown temperature T k , the corresponding DCQT mode thermalizes to the temperature T k at long times, thus enabling us to perform mode-dependent thermometry. Alternatively, if the bath is thermal with an unknown temperature T, a single-mode DCQT suffices for bath-temperature estimation.
Analysis. For simplicity, we restrict our analysis below to a thermal bath with temperature T, probed by an N -level DCQT that is described by the HamiltonianĤðtÞ = _ P N n¼0 ω n ðtÞ n j i n h j with energy spectrum ω n ðtÞ = nωðtÞ for real positive ωðtÞ, and an interaction HamiltonianĤ I ¼Ŝ B. Here, n denotes the level index, and we assumeŜ to be a system operator belonging to a Lie algebra. For a two-level system,Ŝ σ x , the x Pauli operator, while for a harmonic oscillator,Ŝ â þâ y À Á , whereâ,â y denote the annihilation and creation operators, respectively. For a singlemode DCQT, we consider a generic, periodic, diagonal (frequency) modulation: where sðm 0 Þ; cðm 0 Þ are adjustable real constants, corresponding to the m 0 -th frequency harmonic. One can extend this analysis to a multi-mode DCQT probing a multi-mode bath (see Methods).
Here, we focus on controls with τ much smaller than the thermalization time of the system, such that one can adopt the secular approximation, thereby averaging out all rapidly oscillating terms and arriving at a time t ) τ at the steady-statê ρðt ! 1Þ ¼ P N n¼0 ϱ n n j i n h j [34][35][36] . The level populations ϱ n are functions of the bath temperature T, the bath response (correlation) functions Gðω m Þ (see Methods) 37,38 at the m-th sideband frequency ω m ¼ ω 0 þ mΔ, and the modulationdependent m-th sideband weight P m ! 0 (see Methods and Supplementary Notes 2-5). Here, the sidebands m ¼ 0; ± 1 ± 2; arise due to the periodic modulation, the mean frequency ω 0 ¼ R τ 0 ωðtÞdt À Á =τ, and P m 's satisfy the constraint P m P m ¼ 1 [34][35][36] . As can be expected in experiments, we impose an upper bound on the available resources for dynamical control, by considering only frequency modulations such that max m 0 ½ Δ < ω 0 . Unless otherwise stated, we take _ and k B to be unity, and assume the bath response obeys the standard Kubo-Martin-Schwinger condition GðÀωÞ=GðωÞ ¼ exp Àω=T ð Þ 37 . We may infer the bath temperature T from measurements of ϱ n of the thermometer: e.g., through measurement of the average phonon occupation number of a trapped-wavepacket probe (Fig. 1a) 39 , or the fluorescence of a two-level system probe (Fig. 1b). We can assess the effectiveness of our measuring scheme from the QFI as a function of the system and bath parameters. The relative error δT=T is bounded by the minimal achievable error ξ, dictated by the Cramer-Rao bound for optimal positive-operator valued measure, which in this case are the level- Fig. 1 Schematic realization of a quantum thermometer. a Schematic realization of a dynamically controlled quantum thermometer by a motional wavepacket of a state of a cavity, or an ion/atom trapped in an optical lattice potential. The frequency of the trap is periodically varied (black arrow) by modulating the length of the cavity, or the amplitudes of the lasers forming the trap. The wavepacket is in contact with a phonon bath, and relaxes to the corresponding thermal steady-state at long times. b Same, for a spin-chain bath coupled to a two-level system whose resonant frequency is periodically modulated. population measurements. It obeys the relation 6,9,40 δT Here, M denotes the number of measurements, and H ¼ À2 lim ϵ!0 ∂ 2 FðρðT; tÞ;ρðT þ ϵ; tÞÞ=∂ϵ 2 ¼ P N n¼0 j∂ϱ n =∂Tj 2 =ϱ n is the QFI at temperature T 7,8,41,42 , Fðρ 1 ;ρ 2 Þ ¼ Tr being the fidelity betweenρ 1 andρ 2 .
The dynamics of the DCQT, and consequently also the resultant steady-state, depends crucially on the factors P m Gðω m Þ; in general, a large P m Gðω m Þ is beneficial for estimating temperatures T $ ω m (see below, Methods and Supplementary Notes 1-5) 36 . Although we cannot control the bath correlation functions (spectral-response) Gðω m Þ, nevertheless for a given Gðω m Þ, we can tailor the thermometry QFI to our advantage by judicious choices of the periodic control fields, and hence of the corresponding P m 's and ω m 's. For example, a versatile thermometer, which can measure a wide range of temperatures with high-accuracy bound would require a modulation that corresponds to large QFI over a broad range of temperatures, in order to yield ξ ( 1 for finite M. The above scenario can be realized using modulations, which give rise to multiple non-negligible P m 's, or equivalently, large P m Gðω m Þ over a broad range of frequencies. In particular, it is exemplified below using a sinusoidal modulation characterized by a single, or few frequencies. On the other hand, a different modulation would enable the same thermometer to measure temperatures with higher accuracy bound (or, equivalently, larger QFI), but at the expense of the changing of the temperature range over which the DCQT can measure accurately. This can for example be realized using periodic π-pulses, which give rise to only two sidebands, viz., non-negligible P ± 1 34 . Therefore, in contrast to thermometry in the absence of any control, DCQT gives us the possibility of tuning the range and accuracy bound of measurable temperatures. This necessitates an appropriate choice of control parameters (Δ; m 0 ; sðm 0 Þ; cðm 0 Þ) in Eq. (2) depending on the temperatures of interest.
Harmonic-oscillator DCQT for sub-Ohmic baths. As a generic example of DCQT, we consider the probe to be a periodically modulated harmonic oscillator. The interaction Hamiltonian iŝ H I ¼â þâ y À Á B, whereâ (â y ) denotes the annihilation (creation) operator of the DCQT, andB is a bath operator.
We first consider the thermometry of a broad class of bath spectral-response functions 18,43 under the Kubo-Martin-Schwinger condition. Here, γ is a positive constant determining the system-bath coupling strength.
To show the full advantage of our control scheme, we start by focusing on a sub-Ohmic bath spectrum, since it has a non-zero GðωÞ infinitesimally close to ω ¼ 0, even though GðωÞ vanishes at ω ¼ 0 (cf. Eq. (4)), which is ideal for low-temperature thermometry 17 . To simplify the treatment, we take the limit of small γ in Eq. (4) to ensure weak-coupling, s ! 0 to generate small ω min , and ω c ! 1 to realize high-cutoff frequency, and arrive at the nearly flat bath spectrum (NFBS) for ω min ≲ ω ( ω c , with ω min % sω c (see Supplementary Notes 4 and 5). Hence, ω min can be as small as sω c , thus enabling us to probe the bath very close to the absolute zero (see below). The comparison of our results with those obtained for a sub-Ohmic bath spectrum with finite s and ω c is shown in Figs. 2 and 3. The extension of our results to the case of a generic bath spectrum is presented below. Below we choose the control scheme Eq. (2) to have cðm 0 Þ ¼ ω 0 δ m 0 ;0 and sðm 0 Þ ¼ μΔδ m 0 ;1 , so that ωðtÞ is varied sinusoidally: In the limit of weak-modulation (0 μ ( 1), only the sidebands m ¼ 0 and m ¼ ± 1 have significant weights P m in the harmonic (Floquet) expansion of the system response 34 : the mth sideband weight P m , corresponding to the Floquet (harmonic) frequency ω m ¼ ω 0 þ mΔ, falls off rapidly with increasing jmj. The QFI can then be written as (see Supplementary Note 4) Here, H ± 1 are the contributions to the QFI arising from to the m ¼ ± 1 sidebands with P ± 1 ' μ 2 =4, and H 0 includes the m ¼ 0 contribution. The advantages of DCQT are then revealed for a control scheme with Δ chosen such that so that with the average occupation number given by The QFI is then predominantly associated with the H À1 term, with the maximum located at the temperature If higher sidebands (jmj ! 2) are taken into account, the results remain valid, except for a marginal increase of the relative error bound ξ at low temperatures (see Supplementary Notes 2-5). By contrast, in the absence of any control (μ ¼ 0), ϱ n is still given by Eq. (8), with the average phonon occupation number which vanishes in the limit T ( ω 0 . The corresponding QFI is given by which has a single peak at T ¼ T 0 , where T 0 satisfies the equation Fig. 2). One can infer from the above results that a weak sinusoidal modulation (small μ) with appropriately large Δ results in a double-peaked QFI, attaining maxima at T % T À1 and T % T 0 , owing to contributions from H À1 and H 0 , respectively (Fig. 2). This double-peaked QFI signifies the applicability of DCQT for estimating a much broader range of temperatures, viz., in the vicinity of T ¼ T À1 , which can be tuned according to our temperature of interest by tuning ω À1 , and T ¼ T 0 . In comparison, an uncontrolled quantum thermometer can accurately measure temperatures only in the vicinity of T ¼ T 0 for probes characterized by a single energy gap, or at multiple fixed (untunable) temperatures, for probes characterized by highly degenerate multiple energy-levels with arbitrary spacings 12 .
For a sub-Ohmic bath spectrum with a low-frequency edge ω min , the minimum error bound ξðT ¼ T À1 Þ per measurement (M ¼ 1) remains finite and approximately constant at (see where Δ " ω 0 signifies Δ approaching ω 0 from below. Condition (14) yields the maximum possible advantage offered by our control scheme close to the absolute zero (i.e., for T % T À1 ( ω 0 ), and is valid in the regime An error bound ξ, which does not diverge at low temperatures T ( ω 0 (as in standard quantum thermometry), but instead stays constant as the temperature decreases in that range, is deemed to be highly advantageous, since it allows us to measure such low temperatures. We have thus obtained a remarkable result: the advantage offered by our control scheme, expressed by the bound Eq. (14) close to the absolute zero (i.e., for T ( ω 0 ), is maximal for a sub-Ohmic, nearly flat bath spectrum (NFBS) with small, but non-zero, lower edge, such that the bath interacts weakly with the DCQT at all non-zero frequencies. Regime (15) ensures that the dynamics is Markovian (see Methods).
Conditions (14), (15) allow us to achieve thermometry with a high-precision bound for very low temperatures; viz., only a finite number of measurements M ) e 4 =ð4μ 2 Þ ensures a relative error bound, which is constant, i.e., temperature-independent, and has the finite value exp 2 ð Þ=2μ ffiffiffiffiffiffi M p . For given small values of G 0 and ω min , we then arrive at a minimum (limiting) temperature bound that is measurable with the error bound (14) per measurement: In particular, for a sub-Ohmic NFBS (Eq. (5)) coupled very weakly with the thermometer at all frequencies such that G 0 ( ω min ! 0, one has T lim ! 0, thus enabling us (at least under ideal circumstances, i.e., in absence of any other source of error) to accurately measure temperatures very close to the absolute zero. The only penalty for a small G 0 ( ω min is long thermalization time. Yet, even if G 0 corresponds to divergent thermalization times (e.g., for G 0 ; ω min ! 0), one can use optimal control to reach the steady-state at the minimal (quantum speed limit) time 44 : one can initially apply a strong brief pulse to take the thermometer to an optimal state, from which it takes the least time to thermalize, as detailed in ref. 45 . Following this initial pulse, one can periodically modulate the system, as presented here. This behavior is in sharp contrast to that of an unmodulated thermometer, where ξðTÞ diverges for T ! 0, thus precluding any possibility of precise temperature estimation at very low temperatures 17 .  6)), for the following bath spectra: nearly flat bath spectrum (magenta solid curve), sub-Ohmic bath spectrum with s ¼ 0:1; ω c ¼ 100 (black dashed curve) and the same spectra in the absence of control (turquoise dotted curve). Dynamical control reduces the relative error bound significantly, with ξ remaining constant over a broad range of low temperatures T ( ω 0 . The sidebands m ¼ 0; ± 1; ± 2; ± 3 are included in the calculation. Here, the modulation amplitude μ ¼ 0:2, and ω 0 ¼ 1. Inset: Corresponding spectralresponse functions scaled by γ. Fig. 2 Relative error bound and quantum Fisher information. Relative error bound ξ for estimation of bath temperature T (in units of ω 0 ) by a harmonic-oscillator dynamically controlled quantum thermometer (DCQT) under sinusoidal modulation for the following bath spectra: nearly flat bath spectrum (magenta solid curve), sub-Ohmic bath spectrum with s ¼ 0:1; ω c ¼ 100 (black dashed curve) and the same spectra in the absence of control (turquoise dotted curve). For low temperatures in the absence of control, ξ ! 1, thus making it impossible to measure low temperatures as shown. In contrast, our dynamical control scheme reduces ξ to finite values at low temperatures, for nearly flat, as well as sub-Ohmic bath spectra, thus showing the advantage of DCQT. Inset: Quantum Fisher Information (H) as a function of temperature T (in units of ω 0 ) for DCQT under sinusoidal modulation and in the absence of control, for the bath spectra in the main figure (same curve colors). DCQT increases the quantum Fisher information significantly at lower temperatures, giving rise to a peak at T % T À1 (magenta dot), in addition to the peak at T % T 0 (turquoise dot). Here, the modulation amplitude μ ¼ 0:2; ω 0 ¼ 1; Δ ¼ 0:9 (see Eq. (6)), m ¼ 0; ± 1; ± 2; ± 3 and the number of measurements M ¼ 1. The thermalization time $ γ À1 is assumed to be long enough such that the secular approximation is valid.
This novel effect of keeping ξ finite even in close proximity to the absolute zero is a direct consequence of our control scheme, and is unachievable without modulation for any non-zero ω 0 . Extension of the analysis to multipeak QFI is given in the Supplementary Note 6 (also see Fig. 4).
As discussed above, optimal thermometry demands choosing a Δ such that the QFI has a maximum at the temperature regime of interest, which is always possible for small enough ω min (cf. Eq. (16)). However, even for a bath spectrum with small but finite ω min , one can perform low-temperature thermometry with a lowrelative error bound in the regime G 0 ( T < ω min via suboptimal DCQT whose response is peaked at a temperature close to, but larger than ω min . In this case the QFI, albeit not maximal, would be still significantly higher than that of the same thermometer in the absence of dynamical control. For example, in order to measure temperatures of the order T $ 10 À3 ω 0 , ideally one would need ω m $ 10 À3 ω 0 . However, even a suboptimal modulation with ω m $ 10 À2 ω 0 , which leads to QFI with a maximum at T $ ω m $ 10 À2 ω 0 , results in H % 14:3 at T ¼ 10 À3 ω 0 (see inset of Fig. 4). In contrast, H is vanishingly small (i.e., H ( 1=T 2 such that ξ ) 1= ffiffiffiffiffiffi M p ; cf. Eq. (3)) for the same temperature regime in the absence of control, thus exhibiting the advantage of our DCQT even for suboptimal thermometry at T < ω min .
Thermometry of arbitrary bath spectra. The advantage offered by DCQT is not restricted to the specific bath spectra considered above. In fact, the key result of significant improvement in lowtemperature thermometry by dynamical control holds for arbitrary bath spectra satisfying the Kubo-Martin-Schwinger condition (see Eq. (30) of Methods), as long as Gðω > 0Þ is independent of temperature. Choosing a control scheme satisfying (see Eq. (11)) for a positive constant κ of the order of unity, leads to a QFI, which diverges quadratically with temperature: The above result Eq. (18), which is valid for any bath spectra, arises due to ω À1 being vanishingly small (see Eq. (17)). The lowest temperature for which Eq. (18) is valid is given by the condition As in the case of sub-Ohmic NFBS, condition (19) ensures that the thermalization time is long enough for the secular approximation to remain valid. This in turn results in a relative error bound which is not explicitly dependent on the temperature, even for low temperatures T ( ω 0 (see Fig. (3)). Although the relative error bound ξ is not explicitly dependent on temperature, the QFI and consequently ξ still depend on the details of the bath spectral function GðωÞ at ω ¼ ω À1 , which in turn is determined by the temperature through Eq. (17). The maximum QFI in this case is obtained by modulations satisfying the optimal condition P 1 Gðω À1 Þ ) P 0 Gðω 0 Þ; P 1 Gðω 1 Þ: ð21Þ The above condition ensures that the sidebands m≠ À 1 are insignificant, so as to yield a small relative error bound at T $ ω À1 (see Supplementary Note 3). Condition (21) shows that one needs to tailor the control scheme to the bath spectrum at hand. For example, in the case of a bath spectrum characterized by Gðω 0 Þ ) Gðω À1 Þ, one needs to design a modulation with P À1 ) P 0 , in order for our control scheme to be beneficial for estimating temperatures T $ T À1 ( ω 0 . This can be realized using a periodic π-pulse modulation, in which case P ± 1 % 4=π 2 and P m≠ ± 1 % 0 34 .

Discussion
We have shown that by subjecting a generic quantum thermometer to an appropriate dynamical control, we may increase its maximum accuracy bound of measuring a chosen range of temperatures. The class of quantum thermometers considered here relies on the measurement of the energy-level populations ϱ n at the steady-state, as well as knowledge of the resonant frequency ω 0 of the thermometer, the modulation parameters μ and Δ (cf. (6)) and (at least crudely) the bath spectral density GðωÞ (see Supplementary Note 1). It falls in the category of secondary thermometers. The advantage of the proposed DCQT becomes especially apparent at low temperatures, where, for generic bath spectra GðωÞ, its accuracy bound of measuring the bath temperature, quantified by the QFI, is dramatically higher than that of its unmodulated counterpart. Namely, dynamical control allows us to perform low-temperature thermometry with temperature-independent relative error bound, at temperatures above the bound set by Eq. (16). Our proposed control scheme can be tailored according to the bath spectra at hand, in order to maximize the QFI, and determine the number of its peaks and sensitivity ranges, thus making it highly versatile.
In addition to our diagonal HamiltonianĤðtÞ ¼ P N n¼0 ω n ðtÞ n j i n h j, off-diagonal interaction terms of the form P N n;m¼0 ðh n;m n j i m h j þ h:c:Þ may incur non-adiabatic effects, which may cause spurious transitions between the energy levels, even in the absence of any external bath. In order to avoid such excitations one should require ωðtÞ ) h n;m for all t; n; m, thereby ensuring adiabaticity.
The dynamical control studied here has already been realized experimentally, in the context of probing the system-environment coupling spectrum 46 . One can utilize similar dynamical control to realize DCQT in different multilevel systems (see Supplementary Note 1): (i) a harmonic-oscillator DCQT can be implemented by the mode of a cavity of length lðtÞ, whose frequency ωðtÞ / 1=lðtÞ is modulated with the inverse of the cavity length 47 . (ii) A two-level DCQT can be implemented by a qubit whose level spacing is modulated by a time-dependent magnetic field, for example in nuclear magnetic resonance 48 and nitrogen-vacancy (NV)-center 49 experiments. Experimentally, one can probe the thermal steadystateρðt ! 1Þ of the DCQT at large times, after decoupling it from the bath, followed by ceasing the dynamical control (i.e., first setting γ ¼ 0, and then μ ¼ 0). In a harmonic-oscillator thermometer one can then estimate the temperature T through measurement of the mean number of quanta of the thermometer N avg , for example through motional sideband spectroscopy, which relies on the asymmetry between phonon absorption (proportional to N avg ) and phonon emission (proportional to N avg þ 1) 50 . In a two-level NV-center thermometer 51,52 , spin readout using photoluminescence may enable us to measure the N-V temperature.
The thermal energy-level population measurements ϱ n (see Eq. (9)), which constitute the optimal positive-operator valued measure for estimating temperature 40 , are characterized solely by the average number of quanta N avg . Consequently, measurement of N avg gives us the complete knowledge ofρðt ! 1Þ, which in turn enables us to approach the minimum error bound ξ of bathtemperature estimation. However, while DCQT can dramatically reduce the error bound of temperature estimation (see Eqs. (19) and (20)), additional experimental errors may arise from the finite accuracy of measuring Δ, or the energy-level populations. These errors, which depend on the experimental apparatus, may prevent us from reaching the fundamental theoretical bound of Eq. (20) (see Supplementary Note 2).
DCQT can be highly advantageous for thermometry of diverse baths realized by many-body quantum systems in condensed matter and ultracold atomic gases, with spectra and modulations satisfying the optimal condition Eq. (21).
The temperatures measurable with low-relative error bound using our control scheme are presented in Table 1.
In a microwave cavity subjected to sinusoidal modulation with ω 0 À Δ $ 10 x Hz, DCQT results in QFI attaining a peak at temperatures of the order of 10 xÀ12 K, x being real. We thereby open new avenues for the study of cavity quantum electrodynamics and quantum information processing 4 at extremely low temperatures. The caveat noted above is that although the dynamical control can be expected to significantly reduce the error in low-temperature thermometry, however, experimental errors may preclude the attainment of theoretical bound of temperature resolution.
DCQT can be expected to be highly beneficial in baths exhibiting the widely studied 1=f α (α ! 0) noise spectra 43 in vacuum tubes 53 or in thick film resistors 54 . This kind of spectra ensures that for a sinusoidal modulation with small ω À1 , the most dominant contribution arises from the m ¼ À1 sideband, thus enabling us to probe low frequencies with highprecision bound.
An intriguing application of the proposed DCQT concerns experimentally studying the third law of thermodynamics for lowtemperature many-body quantum systems, in the sense of understanding the scaling of the cooling rate with temperature [55][56][57] . In view of predictions that the cooling rate does not vanish as the absolute zero is approached for baths with anomalous dispersion, such as magnon (ferromagnetic) spin chains 58 , high-precision lowtemperature thermometry is imperative.
In many-body quantum systems [59][60][61][62][63] , interactions may keep a system non-equilibrated for a long time after a quench 37 , during which time its different collective modes k may have their own temperatures T k . High-precision multi-mode thermometry over a wide range of temperatures using our DCQT can verify whether different modes have different temperatures, and thus avoid thermalization.
The versatility of our DCQT can be well-suited for simultaneous multi-mode probing of a bath with high-accuracy bound, which can be especially useful for nanometer-scale thermometry in biological systems 64,65 .
The recently discussed need for thermometers with vanishing energy gaps for measuring low-temperatures in many-body quantum systems 17 and in strongly coupled quantum systems 18 , suggests the importance of control schemes capable of tuning the gap of a quantum thermometer to our advantage. Application of our control scheme to thermometers modeled by many-body quantum systems, or to thermometers coupled strongly to the bath, in order to achieve high-precision lowtemperature thermometry, is an interesting question, which we aim to address in the future.
The control scheme presented above may have diverse applications in quantum metrology beyond thermometry. For example, periodic modulation of energy levels in many-body quantum critical systems may be applicable to precise probing of interparticle coupling strengths in these systems 66 .

Methods
Thermalization under periodic modulation. We consider a multi-mode DCQT system with its state ρðtÞ given as a direct product of single-mode states ρ k : ρðtÞ ¼ k ρ k ðtÞ, interacting with a multi-mode bath, again described by the direct product state ρ B ¼ k ρ B k . Here, ρ B k denotes the state of the k-th mode of the bath. For each k, ρ k ðtÞ evolves under the action of a periodic Hamiltonian, satisfying Eq.
(1). The system is coupled to the bath mode through the interaction Hamiltonian where each of the independent k-th mode baths has a spectrum wide enough to give rise to Markovian dynamics; for example, a finite-Q cavity mode, or a finitelifetime phonon mode. In case the above assumption is violated, leading to non-Markovian dynamics 26,37 , the DCQT may be useful for revealing the absence of thermalization. Here, we allow for baths with mode-dependent temperatures T k , which we aim to measure accurately using our DCQT. We sketch below the derivation of the master equation describing the thermalization of a system under periodic control. We refer to refs. [34][35][36][37] for details of the derivation, and ref. 46 Table 1 Illustrative values. Values of T À1 ¼ _ω À1 =4k B , quantum Fisher information HðT ¼ T À1 Þ and relative error bound ξðT ¼ T À1 Þ for ω À1 ¼ ω 0 À Δ $ 10 x Hz, for a harmonic-oscillator dynamically controlled quantum thermometer under sinusoidal modulation probing a nearly flat bath spectrum. Here, x is a real number.

Parameters
Values ω À1 $ 10 x Hz T À1 % 1:9 10 xÀ12 K HðT ¼ T À1 Þ % The time evolution operator for the periodic Hamiltonian Eq. (1) is given bŷ T being the time ordering operator. According to the Floquet theorem, one can decompose the time evolution operator asÛ k ðt; 0Þ ¼P k ðtÞeR k t , whereP k ðt þ τÞ 1 P k ðtÞ andR k is a constant operator. Taking into accountÛ k ð0; 0Þ ¼ 1, and the periodicity ofP k ðtÞ, one obtainsP k ð0Þ ¼ 1 and hencê U k ðτ; 0Þ ¼P k ðτÞeR k τ ¼Pð0ÞeR k τ ¼ eR k τ . One can now identify the operatorR k with an effective HamiltonianĤ k;eff averaged over a period ofĤ k ðtÞ, aŝ U k ðτ; 0Þ ¼ eR k τ ¼: e ÀiĤ k;eff τ ; ð24Þ with eigen energies Ω k;r and eigen states r k j i, through the relation H k;eff ¼ X r Ω k;r k r k j i r k h j: ð25Þ The Fourier componentsŜ ω k ;m of the system operatorŜ k ðtÞ (cf. Eq. (22)) in the interaction picture with respect to the evolution (23), are given aŝ S k ðtÞ ¼Û y k ðt; 0ÞŜ kÛ k ðt; 0Þ ¼ X where Δ ¼ 2π=τ, m are integers, and fω k g is defined as the set of all transition frequencies Ω k;r 0 À Ω k;r between the levels ofĤ k;eff and the operatorsŜ ω k ;m are the mth-harmonic transition operators associated with these levels 36 .
Next, we focus only on the dynamics of the k-th mode of the thermometer and the bath. Under the standard Born-Markov approximation in the weak thermometer-bath coupling limit, we arrive at the master equation 37 d dtρ k ðtÞ ¼ À where Tr B k denotes trace over the bath mode k, andĤ Ik ðtÞ is the k-th mode interaction Hamiltonian in the interaction picture. We now assume that the thermalization time of the thermometer induced by the bath is much longer than τ or ω À1 k;m ω k þ mΔ ð Þ À1 . Consistently, we adopt the secular approximation (SA) to average out the rapidly oscillating terms of the form exp½ ± iðω k;m À ω k 0 ;m 0 Þt at times t ) τ; ω À1 k;m , such that only the terms with k ¼ k 0 ; m ¼ m 0 survive 68 . We then finally arrive at the master equation [34][35][36][37] , which is a weighted sum of Lindbladian superoperators, each valid for system-bath coupling centered at ω k;m :¼ ω k þ mΔ: Here, L ω k ;mρk ðtÞ Â Ã ¼Gðω k þ mΔÞD ω k ;m ½ρ k ðtÞ þ GðÀω k À mΔÞD y ω k ;m ½ρ k ðtÞ; D ω k ;m ½ρ k ðtÞ ¼Ŝ ω k ;mρk ðtÞŜ y ω k ;m À

Data availability
All relevant data are available to any reader upon reasonable request.