Probe Method for Floquet Qubit Spectroscopy

• The probe method is a pump–probe technique that measures absorption between pump-dressed Floquet states in a periodically driven qubit.
• It employs Floquet theory to nonperturbatively treat the strong periodic drive and uses linear response for the weak probe to extract multiphoton resonance conditions.
• This approach enables the determination of quasienergy splittings, effective matrix elements weighted by Bessel functions, and insights into decoherence through resonance linewidths.

The probe method, in the sense developed for Floquet quasienergy spectroscopy, is a pump–probe framework for extracting the quasienergy structure of a periodically driven discrete quantum system by monitoring absorption induced by a weak secondary drive. In the formulation studied for a driven qubit, a strong periodic pump is treated nonperturbatively with Floquet theory, producing pump-dressed quasienergy states, while a weak harmonic probe is treated in linear response and used to induce transitions between those states. The resulting resonance positions map quasienergy differences, and the resonance amplitudes encode matrix elements that depend on the drive orientation, the tunneling term, and the harmonic content of the Floquet states (Silveri et al., 2013).

1. Pump–probe formulation for a driven qubit

The basic system is a two-level Hamiltonian

$$H_0 = \frac{1}{2}(\epsilon \sigma_z + \Delta \sigma_x),$$

where $\epsilon$ is the static energy bias, $\Delta$ is the tunneling splitting, and $\sigma_x,\sigma_z$ are Pauli operators in the diabatic basis. The bare transition frequency is

$\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$

The pump–probe Hamiltonian combines a strong periodic pump at frequency $\Omega$ and a weak harmonic probe at frequency $\omega$:

$$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$

with $a_p \ll A_x, A_z$ (Silveri et al., 2013).

In this parametrization, the longitudinal pump term $A_z \cos(\Omega t)\sigma_z$ modulates the energy bias $\epsilon$0 and generates strong sideband structure in the Floquet spectrum. The transverse pump term $\epsilon$1 drives direct transitions between diabatic states and produces Autler–Townes structure. The weak probe is used to measure absorption between pump-dressed states. A longitudinal probe ($\epsilon$2) does not flip bare states by itself, but in the presence of $\epsilon$3 and pump dressing it acquires an effective transverse component in the Floquet basis; a transverse probe ($\epsilon$4) couples more directly.

The method is therefore not a spectroscopy of bare eigenstates, but of the quasienergy states generated by the strong periodic drive. This is the central conceptual shift: the pump defines the spectral landscape, and the probe reads it out.

2. Floquet construction of quasienergy states

For a pump-periodic Hamiltonian with period $\epsilon$5, the Schrödinger equation admits Floquet solutions

$\epsilon$6

with

$\epsilon$7

and the Floquet eigenproblem

$\epsilon$8

The quasienergies $\epsilon$9 are defined modulo $\Delta$0 (Silveri et al., 2013).

Expanding $\Delta$1 and $\Delta$2 in Fourier harmonics $\Delta$3 converts the problem into a time-independent matrix eigenproblem in Sambe space:

$\Delta$4

where

$\Delta$5

Practical calculations truncate the Fourier index to $\Delta$6 and increase $\Delta$7 until quasienergies and matrix elements converge.

For purely longitudinal modulation, the time dependence can be removed by the unitary rotation

$\Delta$8

Using the Jacobi–Anger expansion

$\Delta$9

one obtains effective $\sigma_x,\sigma_z$0-photon transverse couplings weighted by $\sigma_x,\sigma_z$1. Near an $\sigma_x,\sigma_z$2-photon resonance, the truncated Floquet Hamiltonian in the diabatic basis is

$\sigma_x,\sigma_z$3

with quasienergy splitting

$\sigma_x,\sigma_z$4

This structure exhibits multiphoton dressing with amplitudes set by Bessel functions. Zeros of $\sigma_x,\sigma_z$5 suppress the effective coupling and produce coherent destruction of tunneling. Beyond leading order, off-resonant harmonics shift the resonance condition through dynamic Stark and generalized Bloch–Siegert corrections. In the diabatic basis, the detuning shift is

$\sigma_x,\sigma_z$6

so the resonance condition becomes $\sigma_x,\sigma_z$7 (Silveri et al., 2013).

3. Probe-induced transitions and absorption spectroscopy

The weak probe is described by

$\sigma_x,\sigma_z$8

In the Floquet basis, first-order perturbation theory gives the transition rate

$\sigma_x,\sigma_z$9

where $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$0 denotes the Sambe inner product (Silveri et al., 2013).

This expression states that one probe photon is exchanged while the pump may supply or absorb $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$1 pump quanta. With decoherence, the delta function is replaced by a Lorentzian of linewidth $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$2:

$\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$3

with

$\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$4

The compact golden-rule form used in the paper is

$\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$5

where $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$6 and $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$7 are the steady-state populations of Floquet states.

For probe frequencies between $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$8 and $\hbar \omega_{01} = \sqrt{\epsilon^2 + \Delta^2}.$9, resonances occur when

$\Omega$0

or

$\Omega$1

For a longitudinal probe near an $\Omega$2-photon resonance, the matrix element in the two-level truncation is

$\Omega$3

Thus, a longitudinal probe produces absorption only because $\Omega$4 and pump dressing convert $\Omega$5 into an effective transverse operator in the dressed basis. By contrast, a transverse probe produces more direct Autler–Townes-like transitions.

4. Approximation schemes and spectroscopic regimes

Near an $\Omega$6-photon resonance, the rotating-wave approximation gives

$\Omega$7

The effective multiphoton coupling strength is $\Omega$8. This immediately identifies several regimes: strong resonances when the relevant Bessel factor is large, suppressed resonances at Bessel zeros, and curved avoided crossings once off-resonant shifts are included (Silveri et al., 2013).

For a longitudinal probe, the dressed off-diagonal element implies a probe-induced Rabi rate

$\Omega$9

This links resonance amplitude directly to multiphoton dressing weights.

The longitudinal and transverse pumps generate distinct structures. A transverse pump produces Autler–Townes doublets with splitting approximately set by the pump Rabi strength. A longitudinal pump instead produces Floquet splittings at multiphoton resonances governed by $\omega$0. In the longitudinal case, the resonance condition may be expressed in the diabatic basis through $\omega$1, or in the adiabatic basis through $\omega$2, with additional $\omega$3 or $\omega$4 corrections from off-resonant harmonics.

The paper also gives basis-selection guidance. The diabatic basis is appropriate when $\omega$5 or when strong longitudinal modulation drives Landau–Zener crossings with $\omega$6. The adiabatic basis is preferable when $\omega$7 is comparable to $\omega$8 and $\omega$9 and $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$0. This basis choice affects the number of Fourier blocks required for convergence and the transparency of the analytical approximations.

5. Numerical implementation and dissipative modeling

The numerical procedure begins by specifying $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$1, pump parameters $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$2, and probe parameters $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$3. One then constructs the Sambe-space Floquet matrix, truncates harmonics at $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$4, diagonalizes to obtain $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$5 and $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$6, computes the probe matrix elements $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$7, includes decoherence and nonequilibrium populations, and finally generates absorption maps by scanning $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$8 or the probe frequency $$H(t) = H_0 + A_z \cos(\Omega t)\sigma_z + A_x \cos(\Omega t)\sigma_x + a_p \cos(\omega t)(c_z \sigma_z + c_x \sigma_x),$$9 (Silveri et al., 2013).

The truncation criterion is tied to Bessel weights: for longitudinal driving, $a_p \ll A_x, A_z$0 is chosen so that $a_p \ll A_x, A_z$1 is negligible for $a_p \ll A_x, A_z$2, with $a_p \ll A_x, A_z$3. Convergence is checked by verifying that quasienergies and key matrix elements stabilize as $a_p \ll A_x, A_z$4 increases.

Dissipation is incorporated with a Floquet–Born–Markov treatment. For an Ohmic bath with spectral density

$a_p \ll A_x, A_z$5

at temperature $a_p \ll A_x, A_z$6, the dephasing rate and steady-state population are expressed through Floquet matrix elements

$a_p \ll A_x, A_z$7

The paper gives

$a_p \ll A_x, A_z$8

and

$a_p \ll A_x, A_z$9

Typical parameter ranges considered for Josephson-qubit experiments include, for a Cooper-pair box, $A_z \cos(\Omega t)\sigma_z$0 GHz, $A_z \cos(\Omega t)\sigma_z$1, $A_z \cos(\Omega t)\sigma_z$2, $A_z \cos(\Omega t)\sigma_z$3, and $A_z \cos(\Omega t)\sigma_z$4 mK; and for a flux qubit, $A_z \cos(\Omega t)\sigma_z$5 GHz, $A_z \cos(\Omega t)\sigma_z$6, $A_z \cos(\Omega t)\sigma_z$7, $A_z \cos(\Omega t)\sigma_z$8, and $A_z \cos(\Omega t)\sigma_z$9 mK. In both cases, $\epsilon$00 spans several units to resolve multiphoton structure, while the probe amplitude remains a few percent of $\epsilon$01 to maintain linear response.

6. Experimental signatures, interpretation, and limitations

The method reproduces several characteristic spectroscopic features seen in Josephson-qubit experiments. Resonances follow curves defined by $\epsilon$02 and $\epsilon$03. With small $\epsilon$04 these lines are nearly vertical; with larger $\epsilon$05 they bend and may form closed loops in the $\epsilon$06 plane. Coherent destruction of tunneling appears as discontinuities in resonance lines at zeros of $\epsilon$07. Dynamic Stark and Bloch–Siegert effects appear as bending of resonance curves, especially near $\epsilon$08 and at low photon number (Silveri et al., 2013).

Specific comparisons were made to experiments on a Cooper-pair box by Wilson et al., where calculated absorption reproduced curved resonances and broken lines at Bessel zeros, and to a flux qubit by Izmalkov et al., where calculated absorption showed strong bending and discontinuities consistent with measured phase maps. In the latter case, Stückelberg-type interference can be reinterpreted as a mapping of the underlying quasienergy landscape. The paper also notes that for non-sinusoidal modulation and a quasiperiodic probe, as in Tuorila et al., generalized Floquet methods are required for multiple incommensurate tones.

Several caveats delimit the method. A purely longitudinal probe requires both $\epsilon$09 and pump dressing to generate effective transverse matrix elements. Probe power must remain small to avoid power broadening and probe-induced shifts; otherwise a generalized two-mode Floquet treatment is needed. Large $\epsilon$10 or strong transverse driving demands larger Sambe truncations. The rotating-wave and van Vleck approximations are accurate for $\epsilon$11 and moderate drive amplitudes, but full numerical Floquet calculations are required once $\epsilon$12 or $\epsilon$13 becomes very large. Floquet–Born–Markov modeling further assumes weak system–bath coupling and a secular approximation.

These constraints define the practical scope of the probe method. Within that scope, the method yields quasienergy splittings from resonance positions, Bessel-function dressing weights from resonance amplitudes, and decoherence information from linewidths. This suggests a spectroscopy that is simultaneously structural and dynamical: it resolves the Floquet spectrum of a strongly driven qubit while retaining sensitivity to relaxation, dephasing, and the geometry of probe coupling.