Multi-level quantum noise spectroscopy

System noise identification is crucial to the engineering of robust quantum systems. Although existing quantum noise spectroscopy (QNS) protocols measure an aggregate amount of noise affecting a quantum system, they generally cannot distinguish between the underlying processes that contribute to it. Here, we propose and experimentally validate a spin-locking-based QNS protocol that exploits the multi-level energy structure of a superconducting qubit to achieve two notable advances. First, our protocol extends the spectral range of weakly anharmonic qubit spectrometers beyond the present limitations set by their lack of strong anharmonicity. Second, the additional information gained from probing the higher-excited levels enables us to identify and distinguish contributions from different underlying noise mechanisms.


I. INTRODUCTION
Studying noise sources affecting quantum mechanical systems is of great importance to quantum information processing, quantum sensing applications, and the fundamental understanding of microscopic noise mechanisms [1][2][3][4].Generally, a quantum two-level system -a qubit -is employed as a sensor of noise that arises from the qubit environment including both classical and quantum sources [2,5].By driving the qubit with suitably designed external control fields and measuring its response in the presence of environmental noise, the spectral content of the noise can be extracted [6][7][8][9].Such noise spectroscopy techniques are generally referred to as quantum noise spectroscopy (QNS) protocols.Over the past two decades, QNS protocols have been explored for both pulsed (free-evolution) and continuous (driven-evolution) control schemes and experimentally implemented across many qubit platforms -including diamond nitrogen vacancy centers [10,11], nuclear spins [6], superconducting quantum circuits [12][13][14][15], semiconductor quantum dots [16][17][18][19], and trapped ions [20].Although these protocols have generally focused on Gaussian noise models, a new QNS protocol was recently developed and demonstrated that enables higher-order spectral estimation of non-Gaussian noise in quantum systems [21,22].
Since QNS protocols commonly presume a qubit plat-form, they have generally been developed within a twolevel system approximation, without regard for higher energy levels.As a consequence, despite tremendous progress and successes, QNS protocols have certain limitations (for example, limited bandwidth) when applied to weakly anharmonic qubits such as the transmon [23,24], the gatemon [25,26], or the capacitively shunted flux qubit [27].However, since weakly anharmonic superconducting qubits are among the most promising platforms being considered for realizing quantum information processors [28], noise spectroscopy techniques that incorporate the effects of higher-excited states in these qubits must be developed to further improve their coherence and gate performance.Among existing QNS protocols, the spin-locking approach has been shown to be applicable to both classical and non-classical noise spectra.It is also experimentally advantageous, using a relatively straightforward relaxometry analysis to extract a spectral decomposition of the environmental noise affecting single qubits [13,29], and it has recently been extended to measure the cross-spectra of spatially correlated noise in multi-qubit systems [30].However, despite its efficacy, like most QNS protocols today, it presumes a two-level-system approximation.
In this work, we develop a spin-locking QNS protocol applicable to multi-level systems, and we experimentally validate it using a flux-tunable transmon qubit operated as a multi-level noise sensor.We demonstrate an accurate spectral reconstruction of engineered flux noise over a frequency range 50 MHz to 300 MHz, overcoming the spectral limitations imposed by the sensor's relatively weak anharmonicity of approximately 200 MHz.Furthermore, by measuring the power spectra of dephasing noise acting on both the |0 -|1 and |1 -|2 transitions, we extract and uniquely identify noise contributions from both flux noise and photon shot noise, an attribution that is not possible within solely a two-level approximation.
We consider an externally-driven d-level quantum system (d > 2), which serves as the quantum noise sensor that evolves under the influence of its noisy environment (bath).Throughout this work, we consider only pure dephasing (σ z -type) noise.In the interaction picture with respect to the bath Hamiltonian H B , the joint sensorenvironment system can be described by the Hamiltonian: ω (j)  s + B (j) (t) |j j| + ξ(t)λ (j−1,j) (σ where |j j| is the projector for the j-th level of the multilevel sensor.The sensor eigenenergies are ω (j) s with the ground state energy set to zero, and the B (j) (t) correspond to the time-dependent noise operators that longitudinally couple to the j-th level of the sensor and cause level j to fluctuate in energy.The raising and lowering operators of the sensor are denoted by σ (j−1,j) + ≡ |j j−1| and σ (j−1,j) − ≡ |j − 1 j|, respectively.The external driving field is denoted by ξ(t).We continuously drive the multi-level sensor with a signal where A drive , ω drive , and φ correspond to the amplitude, the frequency and the phase of the driving field, respectively, and we assume φ = 0 without loss of generality.The parameter λ (j−1,j) represents the strength of the |j − 1 -|j transition relative to the |0 -|1 transition with λ (0,1) ≡ 1.
Throughout the main text, we will refer to the reference frame and two-dimensional subspace defined by the j-th spin-locking basis {|+ (j−1,j) , |− (j−1,j) } as the jth spin-locking frame and the j-th spin-locking subspace, respectively.To move to the j-th spin-locking frame, we apply unitary transformations and truncate the Hilbert space of the multi-level sensor into the j-th spin-locking subspace (see detailed derivation in supplementary materials IV).Then, the effective Hamiltonian describing the j-th noise spectrometer is: where σ(j−1,j) z , σ(j−1,j)

+
, and σ(j−1,j) − denote the Pauli Z operator, the raising operator, and the lowering operator of the j-th spin-locked spectrometer, respectively.The operators B(j−1,j) ⊥ (t) and B(j−1,j) (t) denote the noise operators that lead to longitudinal and transverse relaxation of the spectrometer within the j-th spin locking subspace, respectively.They are given as linear combinations of B (j) (t), arising from the level dressing across multiple levels as follows: where we define the noise participation ratio α (j−1,j) ) as a dimensionless factor that quantifies the fraction of the dephasing noise at the k-th level that is transduced (i.e., projected) to transverse (longitudinal) noise of the j-th pair of spin-locked states.The noise participation ratios α (k) (j−1,j) and β (k) (j−1,j) can be estimated by numerically solving for the dressed states in terms of the bare states |j (see supplementary materials IV for details).Note that the sign of the noise participation ratios can be either positive or negative, leading to the possibility for effective constructive and destructive interference between the noise operators B (k) (t).
There are two noteworthy distinctions between a manifestly two-level system and a multi-level system.First, although the splitting energy Ω (j−1,j) between the jth pair of dressed states (the j-th spin-locked states) is predominantly determined by the effective driving energy (λ (j−1,j) A drive ), they are not universally equivalent.For an ideal two-level system within the rotating wave approximation, the Rabi frequency is indeed proportional to the effective driving field via the standard Rabi formula [13,29,30].However, this is not generally the case in a multi-level setting due to additional level repulsion from adjacent dressed states [33].Rather, in the multilevel setting of relevance here, the distinction between Ω (j−1,j) and λ (j−1,j) A drive must be taken into account to yield an accurate estimation of the noise spectrum.
Second, as a consequence of the multi-level dressing, more than two noise operators B (k) (t) generally contribute to the longitudinal relaxation within a given pair of spin-locked states.In the limit where λ (j−1,j) A drive is small compared to the sensor anharmonicities, the Eqs.(4)-( 5) reduce to B(j−1,j) which conform to the standard spin-locking noise spectroscopy protocol for a two-level sensor [13,29,30].However, as the effective drive strength λ (j−1,j) A drive increases, the contribution of peripheral bare states -other than |j − 1 and |j -to the formation of the spin-locked states |+ (j−1,j) and |− (j−1,j) increases.As a result, in the large λ (j−1,j) Ω drive limit, the multi-level dressing transduces the frequency fluctuations of more than two levels to the longitudinal relaxation within the jth spin locking frame.Also, this multi-level effect contributes to the emergence of non-zero transverse relaxation B (j−1,j) (t), terms which would otherwise be absent within a two-level approximation [13,29,30].

B. Noise Spectroscopy Protocol
The multi-level noise spectroscopy protocol introduced here consists of measuring the energy decay rate Γ (j−1,j) 1ρ (i.e., longitudinal relaxation rate) and the polarization σ(j−1,j) z (τ ) in the j-th spin-locking frame, and then uses these quantities to extract the spectral density S(j−1,j) ⊥ of the noise transverse to the spin-locking quantization axis.This in turn can be related to the longitudinal spectral density S (j−1,j) that causes dephasing (i.e., transverse relaxation) in the original, undriven reference frame (the qubit "lab frame" [13]).We begin by preparing the multi-level sensor in the jth spin-locked state |+ (j−1,j) by applying a sequence of resonant π pulses π (0,1) , π (1,2) , • • • π (j−2,j−1) , which act to sequentially excite the sensor from the ground state |0 to state |j − 1 .We then apply a π (j−1,j) /2 pulse along the y-axis of the Bloch sphere, where the north and south poles now correspond to |j −1 and |j , respectively.The pulse acts to rotate the Bloch vector from the south pole to the x-axis, thereby placing the multi-level sensor in the j-th spin-locked state Subsequently, a spin-locking drive with amplitude A drive is applied along the x-axis (collinear with the Bloch vector) at a frequency resonant with the |j −1 -|j transition and for a duration τ .By adiabatically turning on and off the drive, we keep the state of the sensor within the j-th spin-locking subspace.Once the drive is off, a second π (j−1,j) /2 pulse is applied along the y-axis in order to map the spin-locking basis {|+ (j−1,j) , |− (j−1,j) } onto the measurement basis {|j , |j −1 }, and the qubit is then read out.This procedure is then repeated N times to obtain estimates for the probability of being in states {|j and |j − 1 }, which represent the probability of being in states {|+ (j−1,j) and |− (j−1,j) }, respectively.
The above protocol is then repeated as a function of τ in order to measure the longitudinal spin-relaxation decay-function of the j-th spin-locked spectrometer.For each τ , we define a normalized polarization of the spectrometer, where ρ (j,j) (τ ) denotes the population (the probability) of the j-th level.
From the τ -dependence of σ(j−1,j) z (τ ) , we extract both the relaxation rate Γ (ω) at angular frequency ω = Ω (j−1,j) as follows (see supplementary materials V for details): σ(j−1,j) Here, the transverse noise spectrum S(j−1,j) ⊥ (ω) is the Fourier transform of the two-time correlation function of the transverse noise operators acting on the spectrometer: In the following noise spectroscopy measurements, we will record the spin relaxation for the 1'st and 2'nd spinlocked noise spectrometers [Fig.1(b) and (c)].Then, the traces are fit to an exponential decay, allowing us to extract Γ (j−1,j) 1ρ and σ(j−1,j) z (t) | t→∞ .This is repeated for various drive amplitude A drive in order to reconstruct S(j−1,j) ⊥ (ω).For simplicity, we will hereafter refer to the spin-locking noise spectroscopy exploiting the |j − 1 -|j transition as SL (j−1,j) .To validate the protocol, we will perform the spin relaxation experiments both in the presence and in the absence of engineered noise, and distinguish the contributions of T 1 decay and native dephasing noise from the estimation of S(j−1,j) ⊥ (ω) (see supplementary materials VI for details).
SL (0,1) Experiment SL (j−1,j) as a function of drive amplitude A drive .(d) Spectral estimation of the engineered flux noise (Lorentzian, centered at 200 MHz) via SL (0,1) (blue) and SL (1,2) (red) experiments.Two corrections are applied to the estimates under the two-level approximation (crosses), and consist of shifting the frequency (step 1) and adjusting the magnitude (step 2) due to the multi-level dressing.Note that the corrected flux noise spectra for SL (0,1) (blue circles) and SL (1,2) (red circles) are in good agreement with the ideal flux noise PSD (grey filled).Error bars represent ±1 standard deviations.(e)-(f ) Benchmarking the spectral estimation of engineered flux noise ranging from f0 = 50 MHz to f0 = 300 MHz, where f0 corresponds to the center frequency of engineered noise spectra.The different color shades of the PSD estimates correspond to engineered flux noise with different center frequencies.The agreement between the corrected experimental estimates (circles) and the ideal flux noise PSDs (grey filled) indicates that our protocol overcomes the spectral limit imposed by the sensor anharmonicity.
sults in the fluctuation of the qubit effective Josephson energy, thereby fluctuating the energy levels of the transmon sensor.The other noise source arises from photon number fluctuations in the readout resonator.In this case, photon-number fluctuations in the readout resonator cause a photon-number-dependent frequency shift of the energy levels of the sensor.Figure 2(c) shows a reduced measurement schematic.We generate and apply a known level of engineered flux noise and coherent photon shot noise to the qubit, which we then use as a sensor to validate our protocol (see supplementary materials II).We bias the transmon sensor at a flux-sensitive value Φ ext = 0.17 Φ 0 (dashed line in Fig. 2(d)).At this operating point, the energy relaxation times T 1 for |0 -|1 and |1 -|2 transitions are ∼ 58 µs and ∼ 31 µs, respectively.Note that the energy relaxation time for the |1 -|2 transition is approximately half that of the |0 -|1 transition's relaxation time, which is expected for weakly anharmonic systems [37].
To test our protocol, we first demonstrate an accurate spectral reconstruction of engineered flux noise over a range of frequencies -50 MHz to 300 MHz -that are smaller than, comparable to, and larger in magnitude than the transmon anharmonicity (ω )/2π = −207.3MHz.As with the standard spin-locking protocol, the transmon needs to be driven sufficiently strongly to form an energy splitting Ω (j−1,j) between a pair of spin-locked states {|+ (j−1,j) , |− (j−1,j) } at the measurement frequency of interest.However, when the splitting energy is comparable with or larger than the anharmonicity, the driven transmon can no longer be approximated as a two-level system, and the multi-level dressing that results must be carefully incorporated into the analysis to accurately reconstruct the PSD.
The first step in our noise spectrosopy demonstration is to measure the Rabi frequencies Ω (j−1,j) for the |0 -|1 and |1 -|2 transitions as a function of the drive amplitude A drive [Fig.3(a)].For both the Rabi and the SL (j−1,j) measurements, the rising and falling edges of the spin-locking drive envelope is Gaussian-shaped (∝ exp (−t 2 /2σ 2 )) with σ = 12 ns.For a given amplitude, the resulting Rabi frequency for the |j −1 -|j transition is equivalent to the level splitting Ω (j−1,j) between the spin-locked states (|+ (j−1,j) , |− (j−1,j) ).However, recall that the measured Rabi frequencies Ω (j−1,j) begin to deviate from the two-level system approximation (Ω (j−1,j) = λ (j−1,j) A drive ) as the drive amplitude is increased.The discrepancy (Ω − λ (j−1,j) A drive ) is due to the multi-level dressing effect, the influence of other levels beyond the two-level approximation.Alternatively, one can also observe such frequency deviations by using pump-probe spectroscopy techniques (see supplementary materials III) [32,38].As such, the frequency shifts (Ω (j−1,j) − λ (j−1,j) A drive ) due to this multi-level dressing must be taken into account in order to obtain an accurate estimation of the flux noise spectra at frequencies comparable or larger than the anharmonicity.In our experiments, we found that including up to the 4'th excited state [solid curves in Fig. 3(a)] was sufficient to obtain agreement between our numerical simulations and the experimentally observed frequency shifts.
Similarly, we must consider the noise participation of the peripheral bare states introduced through the multilevel dressing effect in order to obtain an accurate spectral esimation at high frequencies.To build intuition, we begin considering the low-frequency (small A drive ) limit, where the longitudinal relaxation for SL (j−1,j) is determined solely by dephasing noise that acts on |j − 1 and |j (Eq.( 6)).Then, in the large A drive limit, the flux noise acting on the peripheral levels also contributes to the longitudinal spin relaxation.Thus, we must use the noise participation ratios α (k) (j−1,j) for each energy level k, including the original two levels and the peripheral levels: where S Φ (ω) denotes the power spectral density of the engineered flux noise at frequency ω/2π, and ∂ω s /∂Φ ext denotes the flux noise sensitivity of the k'th level of the sensor.For our experiment, the values of α (k) (j−1,j) for SL (0,1) and SL (1,2) are numerically estimated and shown in Fig. 3(b) and (c), respectively.We also numerically estimate ∂ω We now reconstruct the spectrum of engineered Lorenzian-distributed flux noise centered at 200 MHz, a frequency comparable to the sensor anharmonicity (see in Fig. 3d).For the sake of comparison, we first plot PSD estimates based on a two-level approximation (crosses).The frequencies of these PSD estimates are shifted by λ (j−1,j) A drive − Ω (j−1,j) from the ideal flux noise spectra (grey shading).We would also conclude (erroneously) that the extracted flux noise PSD amplitude increases as the frequency increases when estimated using the twolevel approximation.In order to estimate the flux noise PSD accurately, the two corrections described above must be applied to the PSD estimates to account for the multi-level dressing effects: Step 1 -a frequency shift; and Step 2 -an amplitude adjustment.Upon applying these corrections, we successfully reconstruct the PSD estimates for the 200 MHz engineered flux noise for both SL (0,1) and SL (1,2) (markers lie on grey region, Fig. 3d).
Using this approach, we benchmark the performance of SL (0,1) and SL (1,2) for a set of the Lorentzian-shaped engineered flux noise spectra which are centered at f 0 = 50, 100, 150, 200, 250, and 300 MHz. Figure 3(e) and (f) compare the ideal noise spectra (grey shading) with the corrected flux noise PSD estimates (circles sitting on the envelope of the grey regions and following a dashed line) measured by SL (0,1) (blue shades) and SL (1,2) (red shades), respectively, and with the uncorrected estimates ("x" shapes following a solid line) that deviate in both the infered frequency and power.The different colors correspond to the different engineered flux noise spectra.The agreement between the corrected PSD estimates and the engineered noise PSDs clearly substantiates the idea that our protocol overcomes the anharmonicity limit of the noise sampling frequency by taking the multi-level dressing effect into account.
We now move on to distinguishing the noise contributions from both engineered flux and photon-shot noise by measuring SL (0,1) and SL (1,2) .Importantly, both noise sources induce frequency fluctuations of the |0 -|1 and |1 -|2 transitions, but with a different and distinguishing relative noise power ( S(1,2) In the case of flux noise, since the degree of transmon anharmonicity is independent of the external magnetic flux threading the transmon loop Φ ext [23], the flux-noiseinduced frequency fluctuations of the |0 -|1 and |1 -|2 transitions are equal: ∂ω (0,1) s Therefore, for low-frequency flux noise that causes dephasing, the relative noise power spectra of SL (1,2)   SL (0,1) is given as: where we have introduced the subscript Φ to indicate flux noise due to Φ ext .In contrast, photon-shot noise induces frequency fluctuations for each level transition that scale with the corresponding effective dispersive strength χ (j−1,j) [29].The photon-number-dependent frequency shift due to photon shot noise affecting the |j − 1 -|j transition is given as δω (j−1,j) s = 2χ (j−1,j) n, where n is the average residual photon number in the resonator.Hence, the relative noise power spectra of SL (1,2) to SL (0,1) for photon shot noise is: where we have introduced the subscript n to indicate photon shot noise.This finding highlights the usefulness of measuring multiple noise spectra in order to deconvolve environmental noise processes.We shall presume that these two independent sources of engineered noise -flux noise and photon shot noise -are the only two sources of transverse noise impacting the spin-locked spectrometers, and so we may define the total transverse noise power spectrum as S(j−1,j) .
We inject flux noise ranging from 1 MHz to 20 MHz with a "box-car" envelope and, simultaneously, coherent photon shot noise from a coherent tone with a frequency that is detuned by ∆/2π = 6.05MHz from the readoutresonator resonance frequency.Fig. 4(c (ω) are similar in magnitude and predominantly match the grey region.Lastly, at 20 MHz, above which no external noise was applied, the data exhibit a discrete jump down to the sensitivity limit of the experiment.This result indicates that we can distinguish the noise contributions from flux and photon shot noise by measuring the two-fold noise spectra S( . More generally, the independent extraction of unknown flux noise and photon shot noise would be performed by measuring S(j−1,j) ⊥ (ω) for a sufficient number of transitions j − 1, j and frequency ranges in order to back out the individual contributions (within certain and appropriate assumptions about the origin and type of noise).

IV. CONCLUSION
In summary, we introduced and experimentally validated a noise spectroscopy protocol that utilizes multiple transitions of a qubit as a quantum sensor of its noise environment.By moving beyond the conventional two-level approximation, our approach overcomes the anharmonicity frequency limit of previous spin-locking approaches.We further show that measuring the noise spectra for multiple transitions enables one to distinguish certain noise sources, such as flux noise and photon shot noise, by leveraging the differing impact of those noise sources on the different transitions.As an example, we measured the two-fold power spectra of dephasing noise acting on the |0 -|1 and |1 -|2 transitions of a transmon, and showed that our protocol can distinguish between externally applied, known, engineered noise contributions from flux noise and photon shot noise.We anticipate that applying this protocol to even higher level transitions (j > 2) of a superconducting qubit sensor will enable one to distinguish other dephasing noise sources, such as charge noise [39].Noise spectroscopy protocols of the type presented here will help provide deeper insight into the nature of decohering noise sources in qubits, such as 1/f noise [3,40], and ultimately lead to a better understanding of their origin.Identifying and mitigating such noise sources will be an important step towards realizing the full promise of quantum sensing and quantum computing.

DATA AVAILABILITY
The data that support the findings of this study may be made available from the corresponding authors upon request and with the permission of the US Government sponsors who funded the work.

COMPETING INTERESTS
Y.S., J.B., A.V., S.G., W.D.O, and Massachusetts Institute of Technology have filed a provisional US patent application related to multi-level quantum noise spectroscopy protocols.

II. MEASUREMENT SETUP A. Cryogenic setup
The experiments were performed in a Leiden CF-450 dilution refrigerator with a base temperature of 20 mK.The device was magnetically shielded with a superconducting can surrounded by a Cryoperm-10 cylinder.There are two lines for input and output; we apply microwave readout tone and measuring the transmission of sample.All attenuators in the cryogenic samples are made by XMA and installed to remove excess thermal photons from higher-temperature stages.We pump the Josephson travelling wave parametric amplifier (JTWPA) to pre-amplify the readout signal at base tempearture [S7].To avoid any back-action of the pump-signal from TWPA, we added a microwave isolator between the samples and the TWPA.On the RF output line, there is a high-electron mobility transistor (HEMT) amplifier (Cryo-1-12 SN508D) thermally connected to the 3 K stage.Two microwave isolators allow for the signal to pass through to the amplifier without being attenuated, while taking all the reflected noise off of the amplifier and dumping it in a 50 Ω termination instead of reaching the sample.

C. Generation of engineered flux noise
We generate engineered flux noise waveforms using the method described in Supplementary Note 3 of Ref. [S5].In all experiments presented in the main text, we consider the Lorentzian-shaped flux-noise PSD as follows:   where P 0 denotes the noise power, ω 0 /2π = f 0 denotes the center frequency of the noise, and ω c /2π = 2 MHz denotes the half-width at half-maximum (HWHM) of the Lorentzian curve.As described in [S5], we discretely sample the noise spectrum by taking harmonics separated by the fundamental frequency, 4 kHz.The noise spectrum is sampled with a high-frequency cutoff (ω 0 /2π + 50 MHz) and a low-frequency cutoff max(0, (ω 0 /2π − 50 MHz)).In each measurement, a new waveform is produced by an arbitrary waveform generator (AWG) and the total number of noise samples is 1,000.Each noise waveform has a duration of 100 µs.

III. PUMP-PROBE SPECTROSCOPY
In Fig. 3(a), we discussed the effect of multi-level dressing (the frequency shift, Ω − λ j A drive ) by describing the deviation of measured Rabi frequency from the one expected in the two-level approximation.Here, we present experimental results of the pump-probe spectroscopy, which is an alternative approach to capture the effect of multilevel dressing.
Before describing the experimental results, we first present the dressed state picture for a driven multi-level system.We consider a multi-level transmon driven by an electromagnetic field, which is tuned to the frequency of the transmon's |0 -|1 transition (Fig. S1).Then, the transmon-photon system can be written as follows: where a time-independent Hamiltonian H 0 represents the sum of the energies of the transmon and the quantized mode of electromagnetic field (photon), and H int (t) denotes the Hamiltonian describing the interaction between the transmon and the photon.We first choose a set of eigenstates of the interaction-free Hamiltonian H 0 as a basis.This basis corresponds to a tensor-product of the transmon states and photon states, |j, n (Fig. S3).In this basis, the transmon-photon system can be represented as multiple ladders of quantized energy levels.Now, we introduce the interaction between the transmon and photon (red double-headed arrow in Fig. S4).In the case of a driven four-level transmon as illustrated in Fig. S4, the four transmon-photon product states |0, n , |1, n − 1 , |2, n − 2 , and |3, n − 3 can be grouped as "the n-excitation manifold" (grey-filled), each having the same number n of excitations in total.The interaction between these product states leads to the formation of dressed states |+ n , |− n , |2 n , and |3 n ; the dressed states correspond to the eigenstates of the Hamiltonian including the interaction.Note that single-photon transitions are allowed between the nearest-neighboring manifolds.Two-photon transitions are available between the next-nearest-neighboring manifolds.
|2 > . Representation of (four-level-) transmon-photon product states.j denotes the excitation level of the transmon and n denotes the number of photons.

→
(2) Two-photon transition ... , which dresses the transmon sensor.We measure single-and two-photon transition frequencies between the dressed states as a function of the pump amplitude (A drive ) by measuring the absorption of a weak probe tone (ω probe ).The splitting between the dressed states |+ and |− corresponds to the Vacuum Rabi splitting and is effectively pushed due the multi-level dressing effect.This effect becomes more significant as the pump amplitude A drive increases.Dotted lines correspond to the simulation data based on the circuit parameters (Table S1).
Assuming that ω drive is larger then any other rate or frequency in this frame, we can perform the rotating wave approximation (RWA), which leads to the Hamiltonian [S7] HRWA (t) = d−1 j=1 ω (j)  s − jω drive + B (j) (t) |j j| + λ (j−1,j) A drive 2 (σ s − ω drive λ (1,2) A drive /2 • • • 0 0 λ (1,2) A drive /2 ω Next, we find a change-of-basis matrix V that diagonalizes the system Hamiltonian HS,RWA , where the eigenenergies E (j) of the j-th dressed states lie on its diagonal entries.Note that the matrix V may not diagonalize the full Hamiltonian HRWA .The frame where the system Hamiltonian HS,RWA is diagnoalized is referred to as the spin locking frame.The full Hamiltonian in the spin locking frame is Since the matrix V is not diagonal in general, the above equation clearly shows that longitudinal noise in the lab frame for a multi-level system leads to both transverse and longitudinal noise in the spin-locking frame.Note that this is not the case for a two-level system.In that case, longitudinal noise in the laboratory frame is fully transduced to transverse noise in the spin-locking frame.
For ω drive = ω , the relevant pair of the dressed states are |+ (j−1,j) , |− (j−1,j) , with energy splitting Ω (j−1,j) = (E (j) − E (j−1) ).This pair of dressed states spans an effective two-level subspace which forms the j-th spin-locking spectrometer.In order to describe the dynamics of the j-th spectrometer, we truncate the d-dimensional Hilbert space of the multi-level system to its two-dimensional subspace.The truncated Hamiltonian H(j−1,j) SL is given by H(j−1,j) | τ →∞ extracted from an experiment performed at a particular Rabi frequency Ω (j−1,j) are related to the transverse noise PSD S(j−1,j) ⊥

Fig. 2 (
Fig.2(a) and (b), the rightmost transmon (blue) operates as a multi-level quantum sensor.The other transmons' modes are far-detuned from the sensor, such that their presence can be neglected (see supplementary materials I).In this work, we focus on two environmental noise channels that couple to the transmon sensor.One noise channel is formed by the inductive coupling of the sensor's SQUID loop to the fluctuating magnetic field in the qubit environment (flux noise).In this case, a fluctuating magnetic flux threading the SQUID loop re-

B.
Room temperature controlOutside of the cryostat, we have all of the control electronics which allow us to apply microwave signals used for the readout and control of the transmon sensor.All the signals are added using microwave power splitters (Marki PD0R413) used in reverse.Pulse envelopes of qubit control signals and readout signals are programmed in Labber software and then uploaded to arbitrary waveform generators (AWG Keysight M3202A).Subsequently, the pulses generated by AWGs are mixed with coherent tone from RF sources (Rohde and Schwarz SGS100A).All components for generating signals are frequency-locked using the 10 MHz reference clock in the Keysight PXIe Chassis M9019A.A detailed schematic is given in Fig S2.

FIG. S1 .
FIG. S1.Qubit spectroscopy as a function of the spectroscopy tone frequency and external magnetic flux threading the SQUID loop of the spectrometer.Solid and dashed curves are obtained by solving the eigen-energies of the circuit Hamiltonian based on the parameters summarized in Table.S1.Note that the dashed orange curve corresponds to the two-photon (|0 -|2 ) transition frequency.
FIG. S2.Electronics and control wiring

FIG. S4 .
FIG. S4.Dressed state representation of the transmon-photon coupled system.The four product states |0, n , |1, n−1 , |2, n−2 , and |3, n − 3 (grey filled) can be grouped as the n-excitation manifold.The interaction between the transmon and photon (red double-headed arrow, Hint) leads to the formation of dressed states.Note that single-photon transitions are allowed between the nearest-neighboring manifolds.Two-photon transitions are available between the next-nearest-neighboring manifolds.