Ultrastrong coupling probed by Coherent Population Transfer

Light-matter interaction, and the understanding of the fundamental physics behind, is the scenario of emerging quantum technologies. Solid state devices allow the exploration of new regimes where ultrastrong coupling strengths are comparable to subsystem energies, and new exotic phenomena like quantum phase transitions and ground-state entanglement occur. While experiments so far provided only spectroscopic evidence of ultrastrong coupling, we propose a new dynamical protocol for detecting virtual photon pairs in the dressed eigenstates. This is the fingerprint of the violated conservation of the number of excitations, which heralds the symmetry broken by ultrastrong coupling. We show that in flux-based superconducting architectures this photon production channel can be coherently amplified by Stimulated Raman Adiabatic Passage, providing a unique tool for an unambiguous dynamical detection of ultrastrong coupling in present day hardware. This protocol could be a benchmark for control of the dynamics of ultrastrong coupling architectures, in view of applications to quantum information and microwave quantum photonics.

Strong coupling between atoms and quantized modes of an electromagnetic cavity 1 provides a fundamental design building block of architectures for quantum technologies 2 . This regime is achieved when the coupling constant g is large enough to overcome the individual decoherence rates of the mode and of the atom, κ γ  g , , and it has been observed in many experimental platforms from standard quantum optical systems 1,3 , to architectures of artificial atoms (AA) [4][5][6] . In such systems small cavity volumes and large AA's dipoles yield values of g up to 1% of the cavity angular frequency ω c and of the AA excitation energy ε. This allows to perform the rotating wave approximation (RWA) yielding the Jaynes-Cummings (JC) model of quantum optics 1 , which describes the dynamics in terms of individual excitations exchanged between atom and mode. This process has been largely exploited for quantum control of AA-cavity architectures 5,7,8 . Recently, fabrication techniques have allowed to go beyond, entering the regime of ultrastrong coupling (USC) 9 , where ω ε ∼ g , c and the RWA breaks down. So far USC has been detected in superconducting [10][11][12][13][14] and semiconducting [15][16][17][18] based architectures essentially via spectroscopic signatures. New physical processes emerge in the USC regime involving multiple photons and many qubits at once 8 . Dynamical detection of population transfer via a USC-specific channel (photon release by decay of the dressed ground state) has been proposed, using spontaneous emission pumping (SEP) [19][20][21] , Raman oscillations 22 . Several dynamical effects have also been predicted, from nonclassical photon statistics 8,23 to Casimir effect 9,24 but despite the large interest, control in time is still an open experimental challenge. Here we show that coherent dynamics amplifies fingerprints of USC in available hardware. Specifically we prove that a protocol similar to Stimulated Raman Adiabatic Passage (STIRAP) in atomic physics [25][26][27] operated in the so called Vee (V) configuration, provides a unique way to attain coherent population transfer via the USC channel.
Demonstration of coherent dynamics in the USC regime would be a benchmark for quantum control, with appealing applications ranging from microwave quantum technologies [28][29][30][31][32] to dynamical control of quantum phase transitions 33,34 .

Results
The quantum Rabi model and STIRAP. USC between a two-level atom (states | 〉 | 〉 g e { , } and energy spitting ε), and a quantized harmonic mode is described by the quantum Rabi model , the full H R comes into play, leading to spectroscopic signatures (see Fig. 1(a)) as the Bloch-Siegert shift observed in ref. 11  , where the only symmetry left implies conservation of the parity of N. In particular the ground state |Φ 〉 0 , which in the JC model is factorized in the zero photon state and the atomic ground state | 〉 g 0 , acquires components with a finite number of photons, corresponding to nonvanishing c 0n for n even and d 0n for n odd. Proposals of dynamical detection of USC 20,22 aim at the detection of such virtual photons 21 by converting them to real ones. To this end one considers a third ancillary atomic level | 〉 u at a lower energcoupled in the experimentally relevanty ε − ′ < 0 19,20,22,35,36 Assuming that the corresponding transitions are far detuned ε ω ′  c and | 〉 u is effectively uncoupled, the Hamiltonian becomes (See the section Methods) In SEP 26 population is pumped from | 〉 u 0 to |Φ 〉 0 and may decay in | 〉 u 2 , due to the finite overlap = 〈 |Φ 〉 ≠ c g : 2 0 02 0 . The process is forbidden in the JC limit, hence detection of this channel, uniquely leaving two photons in the mode, unveils USC 20 . However SEP would have very low yield, since in most of the present implementations of USC architectures c 02 is not large enough. This problem is overcome by a striking evolution of SEP, called Λ-STIRAP 25,27 , a remarkable technique in atomic physics recently extended to the solid state realm [37][38][39][40][41][42] . Being based on quantum interference, STIRAP selectively addresses the target state with ~100% efficiency. We now show how this allows to amplify coherently the USC channel. STIRAP is implemented by using a two-tone control field, are the pump and the Stokes Rabi frequencies, respectively. Under the above assumptions the relevant dynamics involves three levels and it is described by 36 ε ω ε = − ′| 〉〈 | + − ′ | 〉〈 | + |Φ 〉〈Φ | + .
The level scheme for Λ-STIRAP is depicted in Fig. 1(a). The problem is conveniently tackled in a doubly rotating frame 25 The system can be trapped in this dark state despite of being excited by the external fields which interfere destructively. If the system is prepared in | 〉 u 0 , Λ-STIRAP 25,27 is obtained by shining two pulses of width T in the "counterintuitive" sequence (the Stokes pulse Ω t ( ) s before the pump pulse Ω t ( ) p ). Then the adiabatic evolution of the dark state yields ~100% population transfer to | 〉 u 2 . Adiabaticity is crucial for this protocol, and it is attained by using large pulse areas for both fields. Since T is limited by the dephasing time φ T , STIRAP requires appreciable USC mixing c g ( ) 02 to yield a large enough Ω s : if mixing is insufficient population transfer to | 〉 u 2 does not occur, whereas in the USC regime it occurs with nearly unit probability. Therefore detection of = n 2 photons in the cavity at the end of the protocol is a smoking gun for USC.
This simple picture remains valid for the general multilevel dynamics, with driving fields coupled to all the allowed atomic transitions, Fig. 2(a) showing that unit transfer probability is achieved. A key issue is that STIRAP requires g large enough to guarantee adiabaticity for the Stokes pulse, . This (soft) threshold depends linearly on c g ( ) 02 , whereas the efficiency in SEP is much smaller, depending on | | c g ( ) 02 2 (∝g 4 for small g, see Fig. 1(b)). Thus coherence in STIRAP amplifies population transfer by the USC channel.
Few remarks are in order. We have chosen equal peak Rabi frequencies Ω max [ ] t k , to ensure robustness against fluctuations, the property making STIRAP successful in atomic physics 26 (See the section Methods), and checked that leakage from the three-level subspace is negligible, as expected since |Φ 〉 0 is not populated. For small g STIRAP requires a large  s (in our simulations its value would yield e-g Rabi oscillations with Ω = : 600 MHz 0 ) inducing dynamical Stark shifts, and producing a two-photon detuning δ(t) which may suppress population transfer 27,43 . The problem softens in the multilevel structure, where Stark shifts tend to self-compensate, and may be totally eliminated by using appropriately crafted control 43 (see Fig. 2(a)).

Figure 2.
The full Λ-STIRAP dynamics of a AA-harmonic mode system at ε ω = c , is studied, using 19 states, ε ε ′ = 4 (α = 3) and Ω = T 900 0 . Drives are coupled in the experimentally relevant "ladder" configuration g 0 25 and ′ = g 0, using Gaussian pulses (thick lines) and crafted pulses compensating Stark shifts (thin lines). (b) Population P 1 (t) for = g 0 and ′ ≠ g 0 in the RWA (thick black lines) showing that the JC channel alone may led to population transfer. Red curves refer to = .
. . A 13 3, 6 6, 5 3) population transfer occurs due to USC only in the first case, the stray channel interfering destructively for larger g′.
www.nature.com/scientificreports www.nature.com/scientificreports/ Implementation. Implementation of the Λ scheme in real devices faces two major problems. Anticipating the central result of our work. we claim that they can be overcome in a unique way by using STIRAP in the Vee (V) configuration. The first problem is the reliable detection scheme for the two-photons left in the cavity, which is problematic for THz-photons in semiconductors, while GHz-photons in superconducting AA architectures can be detected with circuit-QED measurement technology 8 . Thus multilevel superconducting AAs offer a natural implementation of our proposal. The second problem is the stray (dispersive) coupling of the mode to AA's transitions involving | 〉 u . This has a drastic impact on the reliability of protocols in Λ configuration. We study this point considering an additional stray coupling η ′ = g g between the mode and the AA u-g transition. For the sake of clarity here we review the main results, postponing details of the analysis to the forthcoming section Effect of stray coupling. We denote with |Ψ〉 j the eigenstates of the new Hamiltonian. Insight is gained by perturbation theory in g′: due to the stray coupling the intermediate state |Φ 〉 → |Ψ 〉 0 0 acquires a component onto | 〉 u 1 and | 〉 → |Ψ 〉 u 2 u 2 acquires a component onto | 〉 g 1 . Thus the dipole coupling to the Stokes field is modified: keeping only the corotating g′ term, to lowest order we find s c s 02 2 2 2 This shows that g′ opens a new channel already in the RWA, which allows population transfer to |Ψ 〉 . Therefore the final detection of two photons is not any more a smoking gun for USC. In general the stray coupling g′ interferes destructively with the g-USC channel (see Fig. 2(b)). STIRAP probes selectively the USC channel if and only if the correction in Eq. (7) is so small that a large enough pulse width T can be chosen, allowing adiabatic population transfer by the USC channel only. We obtain a necessary condition by treating in perturbation theory also the counterrotating g (details are discussed later) /ε − 1 is the anharmonicity of the AA spectrum. In this regime the two competing contributions to Ω t ( ) s Eq. (7) are both ∝ g 2 , thus the condition Eq. (8) can be severe, and indeed it is not met by any available design of superconducting AA. In fact in architectures based on the flux qubit exhibiting the largest figures of USC 10-13 , selectivity is lost because the stray coupling is too strong, η  1. The transmon desing 14 , exhibiting the smallest decoherence rates 44,45 , offers smaller η ≈ 1/ 2 , but long coherence times require small anharmonicity,  α | | . 0 1, and again selectivity is lost. Figure 2 shows that it is not possible to select the USC channel even with parameters much more favorable than those of state-of the art devices. We remark that for the same reason all previous proposals of dynamical detection 20,22 are ruled out, i.e. present day hardware does not allow to detect unambiguously USC in the Λ scheme.
Vee STIRAP scheme. The impasse is uniquely overcome by using the V-scheme for STIRAP. We consider a flux qubit, the lowest energy doublet being coupled to a harmonic mode in the USC regime 10-13 , using the AA's second excited state as the ancillary | 〉 u . The system Hamiltonian is AA describes the flux AA, biased by an external magnetic flux Φ = Φ x 0 /2, Φ = h 0 /2e being the flux quantum. This minimizes decoherence 45,46 since H AA is symmetric with respect to fluctuations of Φ x . The corresponding selection rule forbids g-u transitions, thus the full coupling to the mode reads 1 We consider resonant coupling, ε ω = c , and a general control field operated via the magnetic flux (see Fig. 1(a)) are eigenstates of H R , with eigenvalues E 1± , reducing to the JC doublet | 〉 ± | 〉 e g ( 0 1 )/ 2 when the counterrotating term is switched off. population transfer to | 〉 u 2 can occur only in the USC regime. Indeed simulations considering the whole multilevel structure (Fig. 3) show that ~100% population transfer efficiency is achieved if and only if the USC regime is attained, also when the stray coupling is present. This striking success of V-STIRAP, while favored by large anharmonicity (α ≥ 3) and small ratio between the "ladder" matrix elements www.nature.com/scientificreports www.nature.com/scientificreports/ (η ≈ 1/3) of flux-qubits, has a deeper and robust root: the stray JC coupling g′ potentially spoiling USC-selectivity, is not active in the V-scheme. Indeed Fig. 3 shows that USC-selective population transfer is attained with parameters α = .
1 5 and η = 2/3, not satisfying the requirement Eq. (8), and even worse parameters work. Technical details on the JC channel suppression are given in the next subsection. Here we mention that leading corrections to the control Hamiltonian (12) due to the corotating g′ vanish because 〈 |Ψ 〉 = ± nu 0 1 at lowest order in perturbation theory. This makes V-STIRAP unique as a smoking gun for dynamically probing USC, which again is witnessed by the detection of = n 2 photons at the end of the protocol. The probability is approximately the population of |Ψ 〉 2 , for the simulation in Fig. 3). The suppression of the JC channel makes V-STIRAP USC-selective also for smaller α, thus lower microwave frequencies can be used for the driving fields, a key experimental advantage. Another asset of V-STIRAP is that since > ± d g c g ( ) ( ) 1 2 0 2 (see Fig. 1(b)) coupling with the Stokes field is larger. Therefore sufficient adiabaticity is attained with smaller T: this minimize decoherence effects and/or softens the problem of stray dynamical Stark shifts since weaker Stokes fields  s can be used. Notice indeed that in Fig. 3 shorter time scales than in Fig. 2 were used, and that Stark shifts are not apparent.
Few remarks are in order. We took for granted preparation in the state | 〉 u 0 : for = g 0, it is prepared from | 〉 g 0 by standard pulse sequences 47 . But if ≠ g 0 the ground state is |Φ 〉 0 and the above procedure prepares a state which may contain photons. Since the probability ∝ | | c g ( ) n 0 2 is very small for not so large g, while ± d g ( ) 1 2 is large enough to guarantee V-STIRAP population transfer, we expect only some harmless lack of accuracy. Similar arguments ensure that for reasonable parametrization mixing of | 〉 u 0 due to ′ ≠ g 0 is also small. In any case more accurate preparation protocols may be designed to minimize errors.
Concerning decoherence, we know that STIRAP is mainly sensitive to fluctuations in the | 〉 | 〉 u u span{ 0 , 2 } subspace and rather insensitive to other processes 45,48 . Efficient population transfer requires < φ T T , where 1/ φ T is the decoherence rate in the "trapped" subspace, which is approximately the sum of the decay rate κ of the mode and the decay rate γ → u e of | 〉 u in high-quality devices. In such systems these rates are very small, allowing for T up to several dozens of μs. In devices used for USC spectroscopy the mode has a much smaller quality factor, but there should be no fundamental tradeoff between large g and decoherence of the mode alone, allowing for the fabrication of devices exploiting the coherent dynamics in the USC regime. In alternative, with the standard design of high-quality devices large effective couplings ∼ g N g eff could be attained by using few weakly coupled AAs. We checked the dynamics for = N 4 AAs, and we reproduced results of Fig. 3 using half of the value of g 49 . We also checked that the protocol is robust against possible inhomogeneities of the individual couplings of AAs and the possible presence of stray additional modes at multiple frequencies.
We stress that for the detection of USC it would be sufficient to monitor the population of the Fock states | ≥ 〉 n 2 during part of the protocol. Some transient population of the intermediate state is also tolerable, softening the adiabaticity requirement. Decoherence times ∼ φ T T can also be tolerated 48 since at worst efficiency of the USC-selective channel would be larger than 30%. This opens perspectives also for semiconducting structures, where USC-selective Λ-STIRAP could be observed with some progress in techniques for detecting excess THz photons. We finally mention that a diamagnetic term depending on the specific implementation 50 is included in our Hamiltonian by suitably renormalizing the parameters ω c and ′ g g ( , ). Therefore each setup displaying spetroscopic features of USC will also display the STIRAP dynamics described in this work. is studied, using 26 states, ε ε ′ = .
2 5 (α = . www.nature.com/scientificreports www.nature.com/scientificreports/ Effect of stray coupling. Lambda scheme. We now discuss in more detail the effect of stray couplings. In the Λ scheme we add to the undriven Hamiltonian, Eq. (26), the additional stray coupling η ′ = g g between the mode and the u-g transition, which is the relevant one for AAs. Specifically we consider the more general where again the parity of N is conserved thus many amplitudes vanish. If the counterrotating ∼ H c can be treated as a perturbation we can use for |Ψ〉 j the same quantum numbers of the JC eigenstates, τ ≡ j N ( , ) (see the section Methods), the spectrum being of course different (see Fig. 4a).
To fix the ideas we consider ω ε ε = < ′ c , and focus on the limit ε ω ′ | ′ − |  g c , hereafter referred as the dispersive regime (for the stray coupling). In this limit it is convenient to classify |Ψ〉 j with the same quantum numbers of the JC eigenstates for ≠ g 0 and ′ = g 0, namely . Focusing on a simple picture where STIRAP involves only levels resonantly coupled by the drives, and letting the two tone pulse couple the intermediate state | 〉 → |Ψ 〉 g 0 0 with |Ψ 〉 u 0 (pump) and |Ψ 〉 u 2 (Stokes) the physics is described by an effective three-level Hamiltonian. We now need the matrix elements of the control field in this subspace. To this end we must first diagonalize Eqs (13 and 14). The main structure of the amplitudes is captured by diagonalizing ∼ H 0 in the 12-dimensional subspace spanned by the factorized states with ≤ N 4 excitations. This subspace is enough to account for counterrotating terms in leading (first) order, whereas corotating terms are treated exactly. For the pump field we find www.nature.com/scientificreports www.nature.com/scientificreports/ The leading term of the matrix element can be found by noticing that all the amplitudes are of order zero in the small quantity ε ω ′ ′ + g /( ) c c except c u 0 ,1 which is first order, and can be neglected. In particular since in the dispersive regime ≈ f 1 u 0 ,0 the resulting matrix element is ≈ ⁎ c W t ( ) p 0,0 , i.e. in leading order in the stray coupling the pump matrix element in Λ H C is unaffected. Instead for the Stokes field we find substantial differences. The matrix element in the 12-dimensional subspace is C u s u u u 0 2 0,2 2 ,2 0 ,1 2 ,1 0 ,0 2 ,0 The first term is the Stokes matrix element in Λ H C modified by the stray coupling g′. Indeed in the dispersive regime we can approximate ≈ f 1 u 2 ,2 , and we recover the expression in Λ H C . The two extra terms appearing in Eq. (17) are due to the stray coupling only: the second term is due to the JC part g′, whereas the third depends on the counterrotating part and vanishes if ′ = g 0 c . These extra terms are important since in the physical case = g g c and ′ = ′ g g c they may be of the same order of the first one, modifying substantially the matrix element. To clarify this point we notice that in the dispersive regime for the stray coupling we also have ≈ where the last line is obtained by estimating extra terms by perturbation theory in g′ and ′ g c , which gives in leading order Notice that our definitions imply that ⁎ c 0,2 and c u 2 ,1 have different signs therefore the extra terms due to stray coupling interfere destructively with the amplitude due to the counterrotating g c marking USC. In particular this happens even if the stray coupling is purely corotating, i.e. for ′ ≠ g 0 and ′ = g 0 c . This remarkable illustrative case corresponds to Eq. (7), showing that a non-vanishing g′ yields a nonzero Stokes matrix element even if = g 0 c . This is sufficient to determine population transfer to the two-photon target state, if the matrix element the is large enough to guarantee the global adiabaticity condition for STIRAP, . This picture describes also the physics beyond perturbation theory. First of all Fig. 5 shows that the approximation used in this subsection to evaluate the dipole matrix elements are accurate and that the error we make omitting the stray g′ corotating term is relevant for the Λ configuration. The consequences for the dynamics are shown in Fig. 2b (black curves) for system of 30 coupled eigenstates, also accounting for the general coupling structure of the two-tone driving field W(t).
The necessary condition for detecting unambiguously the USC channel, Eq. (8) is found by arguing that this latter must satisfy the global adiabaticity condition, . Equation (8) is obtained by using the perturbative result (25) for c 0,2 , and letting = g g c . Destructive interference of the = g g c and the g′ channels is illustrated in Fig. 2b (red curves), where it is seen that this physical picture holds beyond perturbation theory. . The remarkable fact is that the matrix element vanishes for both ′ → g g , 0 c c , therefore to this level of accuracy no population transfer may occur due to the corotating stray coupling g′. This picture describes also the physics beyond perturbation theory. Figure 5 shows that the the stray g′ corotating term does not contribute in a relevant way to the dipole matrix elements entering the Vee configuration. The dynamics www.nature.com/scientificreports www.nature.com/scientificreports/ illustrated in Fig. 3, where the whole structure of the drive is accounted for, shows that stray population transfer due to the corotating g′ is suppressed by six orders of magnitude even at an accuracy level beyond perturbation theory. Extra contribution is possibly due to the stray counterrotating ′ g c , which in the physical situation of Fig. 3 is also very small. Summing up corotating couplings do not produce population transfer, whose observation marks unambiguously the emergence of USC.

Discussion
In conclusion, we propose the dynamical detection of the USC regime of light-matter interaction using an ancillary atomic level as a probe. The opening of a USC-specific channel for population transfer is witnessed by detecting two-photons in the harmonic mode. This process, coherently amplified by STIRAP, marks the symmetry broken by the USC counterrotating g term in H R Eq. (1), which determines the violation of the conservation of N Relying on coherent dynamics, the experiment we propose would be a benchmark for adiabatic quantum control of circuit QED architectures 36,51 in the USC regime. STIRAP is known to be superior to other protocols [25][26][27] in the Λ scheme, as SEP 20 or Raman oscillations 22 . What makes it unique is the possibility to operate in V-configuration, which is resilient to the presence of stray AA-mode couplings, inevitable in three-level USC solid state architectures. Flux qubits, offering the largest g/ω > 1 c fabricated so far, also meet all the quantum hardware requirements for the experiment. Notice that these expressions are perturbative in g c but nonperturbative in g, the JC model being recovered for → g 0 c . The nonzero overlap with the > N 0 states marks the emergence of USC. At leading order ∝ c g g c 02 therefore in the physical case = g g c the amplitude ∝ c g 02 2 (see Fig. 1(b) in the main text). The fact that STIRAP depends on Ω ∝ | | c s 02 while SEP depends on | | ∝ c g 02 2 4 , which is much smaller, is one of the assets of coherent amplification.