# Inversion of qubit energy levels in qubit-oscillator circuits in the deep-strong-coupling regime

###### Abstract

We report on experimentally measured light shifts of superconducting flux qubits deep-strongly-coupled to an LC oscillator, where the coupling constant is comparable to the qubit’s transition frequency and the oscillator’s resonance frequency. By using two-tone spectroscopy, the energies of the six-lowest levels of the coupled circuits are determined. We find a huge Lamb shift that exceeds 90% of the bare qubit frequencies and inversion of the qubits’ ground and excited states when there is a finite number of photons in the oscillator. Our experimental results agree with theoretical predictions based on the quantum Rabi model.

According to quantum theory, the vacuum electromagnetic field has ”half photon” fluctuations, which cause several physical phenomena such as the Lamb shift Lamb and Retherford (1947). A cavity can enhance the interaction between the atom and the electromagnetic field inside the cavity, and more precise measurements on the influence of the vacuum becomes possible. A cavity/circuit-quantum electrodynamics (QED) system in the weak and strong coupling regimes is well described by the Jaynes-Cummings Hamiltonian Blais et al. (2004); Walls and Milburn (2007). In the strong coupling regime, when the cavity’s resonance frequency is on resonance with the atom’s transiton frequency , the vacuum Rabi splitting Thompson et al. (1992); Wallraff et al. (2004); Kato and Aoki (2015) and oscillation Brune et al. (1996); Johansson et al. (2006) have been observed. In the off-resonance case, the Lamb shift Heinzen and Feld (1987); Brune et al. (1994); Fragner et al. (2008) caused by the vacuum fluctuations, and the ac-Stark shift proportional to the photon number in the cavity, were observed Brune et al. (1994); Schuster et al. (2005, 2007); Fragner et al. (2008). In the so-called ultrastrong coupling regime Niemczyk et al. (2010); Forn-Diaz et al. (2010), where the coupling constant becomes around 10% of and , and the deep-strong-coupling regime Yoshihara et al. (2017a, b), where is comparable or larger than and , the rotating wave approximation used in the Jaynes-Cummings Hamiltonian breaks down and the system should be described by the quantum Rabi model Hamiltonian Rabi (1937); Jaynes and Cummings (1963); Braak (2011). In these regimes, the light shifts of an atom could non-monotonously change as increases, and the amount of the shift is not proportional to the photon number in the cavity Ashhab and Nori (2010). In this work, to study the light shift in the case of , we use qubit-oscillator circuits which comprises a superconducting flux qubit and an LC oscillator inductively coupled to each other by sharing a loop of Josephson junctions that serves as a coupler. By using two-tone spectroscopy Fink et al. (2008), energies of the six lowest energy eigenstates were measured, and photon-number-dependent qubit frequencies were evaluated. We find Lamb shifts over 90% of the bare qubit frequency and inversion of the qubit’s ground and the excited states when there is a finite number of photons in the oscillator.

The circuits that we use for this work can be described as a composite system that comprises a flux qubit Mooij et al. (1999) inductively coupled to a lumped-element LC oscillator as shown in Fig. 1(a). The circuit design is similar to those of Refs. Yoshihara et al. (2017a, b). The qubit-oscillator circuit is described by the Hamiltonian

(1) |

The first two terms represent the energy of the flux qubit written in the basis of two states with persistent currents flowing in opposite directions around the qubit loop Mooij et al. (1999), and . are Pauli operators. The parameters and are the tunnel splitting and the energy bias between and , where can be controlled by the flux bias through the qubit loop . The third term in Eq. (1) represents the energy of the LC oscillator, where [see Fig. 1(a)] is the resonance frequency, and and are the creation and annihiration operators, respectively. The fourth term in Eq. (1) represents the coupling energy.

At , the Hamiltonian in Eq. (1) is equivalent to that of the quantum Rabi model:

(2) |

Note that the qubit part of the Hamiltonian is now written in the qubit’s energy eigenbasis. When , the energy eigenstates are product states:

(3) |

where and are the qubit’s ground and excited states; the bonding and anti-bonding states of and , and is the -photon Fock state of the oscillator. Here, we name the energy eigenstates ( g, e).

For nonzero values of , the energy eigenstates of cannot be written by Eq. (3) any more. However, in the limit , the energy eigenstates are well described by Schrödinger-cat-like entangled states between persistent-current states of the qubit and displaced Fock states of the oscillator Ashhab and Nori (2010):

(4) |

Here, is the displacement operator, and is the amount of displacement. Note that the displaced vacuum state is the coherent state . Note also that Eq. (4) is valid for any value of and reduces to Eq. (3) when . The photon-number-dependent qubit frequency is defined as the energy difference between the energy eigenstates and , and can be calculated as

(5) | |||||

Here, are Laguerre polynomials; , , , and so on. The difference between and the bare qubit frequency can be considered as the -photon ac-Stark shifts . In particular, is referred to as the Lamb shift. Note that the Bloch-Siegert shift Bloch and Siegert (1940); Cohen-Tannoudji et al. (), the contribution from the counter-rotating terms, is included in the -photon ac-Stark shifts. Since , considerable Lamb shift is expected when becomes comparable to . Considering that has zeros, i.e. points where is equal to zero, also has zeros, and hence, become both positive and negative. In other words, the qubit’s ground and excited states exchange their roles everytime when due to -photon ac-Stark shift. Remarkably, is reduced from by a factor that is determined by the overlap integral of the interaction-caused displaced -photon Fock states of the oscillator in opposite directions as seen in the second line of Eq. (5). By considering the symmetry of , the expressions of energy eigenstates can be applied to any parameter sets of SM (). Also, it is found that Eq. (5) can describe the qualitative behavior of as long as . These two facts allow us to uniquely determine including its sign for all the parameter sets in this work SM ().

To determine parameters of the qubit-oscillator circuits (, , and ), spectroscopy was performed by measuring the transmission spectrum through the transmission line that is inductively coupled to the LC oscillator [Fig. 1(a)]. In total, nine sets of parameters (A–I) in five circuits were evaluated. The shared inductance of the circuit for set A is a superconducting lead, while that of the circuits for sets B–I is a loop of Josephson junctions, where eight flux bias points in four circuits were used. Therefore, much larger is expected for sets B–I. When the frequency of the probe signal matches the frequency of a transition , where stands for the ground state and with stands for the th excited state of the coupled circuit, the transmission amplitude decreases, provided that the transition matrix element is not 0. Note that for nonzero values of , we have labeled the energy eigenstates using a single integer instead of the label used above. Figure 2 shows the amplitudes of the transmission spectra for sets A and H. Here, is the probe frequency, and and are respectively measured and background transmissions SM ().

The parameters are obtained from fitting the experimentally measured resonance frequencies to those numerically calculated by diagonalizing with , and treated as fitting parameters. In Fig. 2, the right panels show the calculated transition frequencies superimposed on the measured spectra. In Fig. 2(a), one can see the splitting of and transition frequencies around , known as the qubit-state-dependent frequency shifts of the oscillator. From the fitting, the parameters are obtained to be GHz, GHz, and GHz. The spectrum shown in Fig. 2(b) looks qualitatively different from that in (a) as discussed in Ref. Yoshihara et al. (2017b). The parameters are obtained to be GHz, GHz, and GHz. Here, is larger than both and , indicating that the circuit is in the deep-strong-coupling regime Ashhab and Nori (2010). The parameters from all the sets are summarized in Table 1.

A | 1.246 | 6.365 | 0.42 | 1.236 | 1.215 | |

(1.235) | (1.213) | |||||

B | 1.01 | 6.296 | 5.41 | 0.233 | -0.452 | -0.13 |

(0.229) | (-0.448) | (-0.123) | ||||

C | 0.92 | 6.288 | 5.59 | 0.193 | -0.412 | -0.062 |

(0.189) | (-0.410) | (-0.059) | ||||

D | 3.93 | 5.282 | 5.28 | 0.54 | -1.512 | 0.56 |

(0.539) | (-1.503) | (0.624) | ||||

E | 4.88 | 5.230 | 5.37 | 0.607 | -1.746 | 0.906 |

(0.607) | (-1.741) | (1.018) | ||||

F | 4.71 | 5.220 | 5.46 | 0.538 | -1.642 | 1.005 |

(0.542) | (-1.641) | (1.087) | ||||

G | 3.53 | 5.263 | 5.58 | 0.375 | -1.255 | 0.8 |

(0.379) | (-1.244) | (0.834) | ||||

H | 1.68 | 6.345 | 7.27 | 0.127 | -0.518 | 0.5 |

(0.122) | (-0.514) | (0.523) | ||||

I | 1.61 | 6.335 | 7.48 | 0.099 | -0.458 | 0.493 |

(0.099) | (-0.451) | (0.532) |

To obtain the photon-number-dependent qubit frequency (, 1, 2), at least five transition frequencies out of seven allowed transitions [Fig. 1(b)] are necessary. However, in each spectrum at , we see only two signals at frequencies and corresponding to the transitions and , respectively. Since the base temperature of the dilution fridge is around 20 mK, only the two lowest energy levels and are populated, and should be taken into account as initial states. Moreover, the transition cannot be measured directly since the transition frequency is below the frequency range of the measurement setup (4 GHz to 8 GHz).

To access transitions other than and , two-tone spectroscopy was used, where a drive signal with frequency is applied while the transmission of a probe signal with frequency around the frequencies or is measured. When is equal to the frequency of an allowed transition involving at least one of these four states, the Autler-Townes splitting Autler and Townes (1955) takes place, which results in the increase of the probe transmission. Figure 3 shows the measured two-tone transmission spectra from set H. An avoided crossing between a horizontal line and a diagonal line is observed in each panel. Interestingly, the slope of the diagonal line is for panels (a) and (b), and 1 for panel (c), which indicate that for panels (a) and (b), while for panel (c). Together with the frequencies numerically calculated from , the corresponding transitions are determined as shown in the right-hand side of each spectrum. The spectrum in panel (c) demonstrates that the energy of is higher than that of and hence is negative. In other words, the qubit’s energy levels are inverted.

Moreover, from these three two-tone transmission spectra, five transition frequencies, , , , , and , can be evaluated; For panel (a), the holizontal line represents the condition of one-photon resonance, , whereas the diagonal line represents the condition of two-photon resonance, . For panel (b), similarly, the holizontal line is and the diagonal line is . For panel (c), the holizontal line is and the diagonal line is . From these five transition frequencies, all the eigenenergies up to the fifth-excited state can be determined. One thing is worth emphasizing here. In the two-tone spectroscopy of a deep-strongly-coupled qubit-oscillator circuit, the states of the qubit are doubly dressed: one is a conventional dressing by a classical strong drive field while the other is in the quantum regime due to deep-strong coupling to the oscillator, where the oscillator’s states are displaced. The experimental results demonstrate that the two kinds of dressed states coexist.

In Fig. 4, normalized photon-number-dependent qubit frequencies ( = 0, 1, 2) obtained from the two-tone transmission spectra are plotted for nine sets of parameters. The axis is the normalized coupling constant . The parameters , , and are obtained from the transmission spectra. These results demonstrate huge Lamb shifts , some of them exceeds 90% of the bare qubit frequencies . These results also demonstrate that 1-photon and 2-photon ac-Stark shifts are so large that and change their signs depending on . The solid lines are theoretically predicted values given by Eq. (5). Table 1 shows a comparison between the measured and the numerically calculated using and the parameters , , and . In many circuits, the measured is smaller than the numerically calculated one, while the agreement of and are good, where the deviations are at most 10 MHz. As discussed in SM (), the numerically calculated in the range is larger than given by Eq. (5) and hence the agreement of in Fig. 4 is a coincidence. In this way, our results can be used to check how well the flux qubit-LC oscillator circuits realize a system that is described by the quantum Rabi model Hamiltonian. A possible source of the deviation in is higher energy levels of the flux qubit. As discussed in Ref. Yoshihara et al. (2017b), the second or higher excited states can shift energy levels of the qubit-oscillator circuit, even though there is energy difference at least 20 GHz between the first and the second excited states. Consideration of the higher energy levels are necessary to identify the origin of the deviation in .

In conclusion, we have studied strongly-coupled and deep-strongly-coupled flux qubit-LC oscillator circuits. By using two-tone spectroscopy, frequencies of the six lowest energy eigenstates are measured, and photon-number-dependent qubit frequencies were evaluated. We find Lamb shifts over 90% of the bare qubit frequency, and inversions of the qubit’s ground and excited states caused by the 1-photon and 2-photon ac-Stark shifts. The results agree with the quantum Rabi model, giving further support to the validity of the quantum Rabi model in describing these circuits in the deep-strong-coupling regime.

###### Acknowledgements.

We thank Masahiro Takeoka for stimulating discussions. This work was supported by the Scientific Research (S) Grant No. JP25220601 by the Japanese Society for the Promotion of Science (JSPS), and JST CREST Grant Number JPMJCR1775, Japan.## Supplemental Material

## S1 symmetry of quantum Rabi model and state assignment from the spectra

The parity operator of the qubit-oscillator system is defined as , where, and are respectively the parity operators of the qubit and the oscillator. The parities of states and operators are defined as follows; For the eigen states, the parity is “” when the eigen value is 1, and the parity is “” when the eigen value is . The parity of an operator is “” when , and is “” when . The parity symmetry in the states and operators that appear in the quantum Rabi model Hamiltonian

(S1) |

is summarized in Table SI. Because both and have negative parity, their product has positive parity, meaning that all three terms in have positive parity. Therefore, , and hence, the energy eigenstates are also eigenstates of . Note that this property does not depend on the values of , , and .

Although the energy eigenstates of cannot be described simply for arbitrary values of , , and , the symmetry allows to define energy eigenstates of as , where (, e) indicate that the qubit is in “g” the ground or “e” the excited state and the oscillator is in the -photon Fock state, Since the parity of is “”, the transition matrix elements is non-zero when the parities of the energy eigenstates and are different, and is zero when the parities are same.

From the transition frequencies alone, the energy eigenstates cannot be determined uniquely. However, by using the parity symmetry discussed above, energy eigenstates are recursively determined as long as in the following way. (i) The ground and the first excited states of a coupled circuit are respectively and , since there is no energy-level crossing between them. (ii) Between the (2)th and th excited states (), the state having nonzero transition matrix element with the state is . In this way, photon-number-dependent qubit frequency can be uniquely defined for all the parameter sets in this work.

parity | ||
---|---|---|

qubit state | ||

photon state | even number | odd number |

qubit operator | ||

photon operator |

## S2 background transmission amplitude

The amplitudes of the measured transmission spectra are fitted by the following formula:

(S2) |

where

(S3) |

and we assumed that a background is independent of energy bias and is written by a polynomial of the probe photon frequency . Eq. (S3) can be applied to a transmission line that is inductively and capacitively coupled to an LC oscillator Khalil et al. (2012), where, is the total quality factor of the resonator, is the external quality factor due to the coupling to the transmission line, and is a phase factor that accounts for the asymmetry of the resonance lineshape. As written in the main text, becomes larger than 1 depending on the value value of .

## S3 numerically calculated

In Fig. S1, normalized photon-number-dependent qubit frequencies obtained from the two-tone spectroscopies are plotted in open stars for set E, which has largest value of . The solid lines are theoretically predicted values in the limit :

(S4) |

which is also given in the main text. The dotted lines are numerically calculated values from and the parameter . Although there are clear deviations in smaller values of , the qualitative behaviors of solid and dotted lines are similar. Interestingly, the blue open star (measured ) is on the solid line rather than the dotted line, where the latter is expected to give more accurate prediction. The numerically calculated in the range is larger than given by Eq. (S4) and hence the agreement of the blue open star and the solid line is a coincidence.

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