Large collective Lamb shift of two distant superconducting artificial atoms

TL;DR

This paper demonstrates a significant collective Lamb shift in two distant superconducting qubits, achieved by suppressing decay with a mirror, revealing strong virtual photon-mediated interactions.

Contribution

The study reports the first observation of a large collective Lamb shift in distant superconducting qubits, surpassing linewidths and showing interaction without decay into the transmission line.

Findings

Collective Lamb shift of 0.8% of qubit frequency observed

Qubits interact via transmission line even without decay into it

Decay suppression enhances visibility of the Lamb shift

Abstract

Virtual photons can mediate interaction between atoms, resulting in an energy shift known as a collective Lamb shift. Observing the collective Lamb shift is challenging, since it can be obscured by radiative decay and direct atom-atom interactions. Here, we place two superconducting qubits in a transmission line terminated by a mirror, which suppresses decay. We measure a collective Lamb shift reaching 0.8% of the qubit transition frequency and exceeding the transition linewidth. We also show that the qubits can interact via the transmission line even if one of them does not decay into it.

Tables

Table 1. Table 1: Extracted and derived qubit parameters. We extract ω 10 subscript 𝜔 10 \omega_{10} , Γ Γ \Gamma and γ 𝛾 \gamma from fitting the spectroscopic magnitude and phase data according to Ref.  Hoi  et al. ( 2013 ) . Note that the effective relaxation rate Γ Γ \Gamma at an antinode is twice what the relaxation rate would be in an open transmission line. The velocity v 𝑣 v is extracted by finding multiple nodes of the field for Qubit 1  Sup . From the two-tone spectroscopy, we extract the anharmonicity, which approximately equals the charging energy E C subscript 𝐸 𝐶 E_{C} of the transmon qubits. We calculate γ ϕ subscript 𝛾 italic-ϕ \gamma_{\phi} from Γ Γ \Gamma and γ 𝛾 \gamma , and the ratio β = C c / C Σ 𝛽 subscript 𝐶 c subscript 𝐶 Σ \beta=C_{\rm c}/C_{\Sigma} between the coupling capacitance C c subscript 𝐶 c C_{\rm c} to the transmission line and the qubit capacitance C Σ subscript 𝐶 Σ C_{\Sigma} from Γ Γ \Gamma and E C subscript 𝐸 𝐶 E_{C} .

Qubit	$E_C/h$ [GHz]	${\omega }_{10}/2\pi$ [GHz]	$\Gamma /2\pi$ [MHz]	${\gamma }_{\varphi }/2\pi$ [MHz]	$\gamma /2\pi$ [MHz]	$\beta$	$v$ [${10}^8$m/s]
$Q1$	$0.324$	(Antinode) $4.068$	$27.18$	$2.15$	$15.74$	$0.717$	$0.8948$
$Q2$	$0.406$	(Antinode) $4.746$	$28.03$	$2.785$	$16.8$	$0.696$

Equations

$$2\Delta ={\Gamma }_0\left\{\sin \left[\frac{{\omega }_{10}}{v}(x_1+x_2)\right]+\sin \left[\frac{{\omega }_{10}}{v}\left|x_1-x_2\right|\right]\right\},$$

Full text

Introduction.

In 1947, when attempting to pinpoint the fine structure of the hydrogen atom, Lamb and Retherford Lamb and Retherford (1947) discovered a small energy difference between the levels $2S_{1/2}$ and $2P_{1/2}$, which were thought to be degenerate according to Dirac’s theory of electrons. This energy difference between the two levels can be understood when vacuum fluctuations are included in the picture, as was verified later by self-energy calculations in the framework of quantum field theory Bethe (1947); Welton (1948); Cohen-Tannoudji (1986). Briefly put, a hydrogen atom will emit photons which are instantaneously reabsorbed; while these “virtual” photons are not detectable by themselves, they leave their traces in the Lamb shift.

The hydrogen atoms that Lamb and Retherford used for their experiment were obtained from molecular hydrogen through tungsten catalyzation. Since the conversion rate for this process was very low, the $2S_{1/2}$ level was only populated in a few atoms. Hence, the observable effects of virtual photon processes were limited to self-interaction; exchanges of virtual photons between atoms could not be detected. However, it was later realized that atom-atom interaction mediated by virtual photons also gives rise to an energy shift, referred to as a collective, or cooperative, Lamb shift Fain (1959); Lehmberg (1970); Arecchi and Kim (1970); Friedberg et al. (1973); Scully and Svidzinsky (2010). The atom-atom interaction also underpins the collective decay known as Dicke superradiance Dicke (1954); Shammah et al. (2018).

There are several obstacles impeding the experimental observation of the collective Lamb shift. The shift can be enhanced by using many atoms, but, if these atoms are too close together, direct atom-atom interactions (not via virtual photons) can obscure the effect. Furthermore, the interaction giving rise to the collective Lamb shift is relatively weak in three dimensions, and the shift can also be hidden by the radiative linewidth (e.g., due to the collective decay). Despite these obstacles, there have been a few experimental demonstrations of collective Lamb shifts: in xenon gas Garrett et al. (1990), iron nuclei Röhlsberger et al. (2010), rubidium vapor Keaveney et al. (2012), strontium ions Meir et al. (2014), cold rubidium atoms Roof et al. (2016), and potassium vapor Peyrot et al. (2018). Mostly, these experiments used developments in atomic trapping and cooling Lukin (2003) that have enabled higher densities of atomic ensembles, leading to a strong coupling between atomic condensates and cavity fields Brennecke et al. (2007); Colombe et al. (2007). An improved theoretical understanding Scully et al. (2006); Svidzinsky and Chang (2008); Scully (2009) of collective Dicke states also aided some of the experiments.

With the single exception of Ref. Meir et al. (2014), these previous experiments all required a large number of atoms to demonstrate a collective Lamb shift. The experiment of Ref. Meir et al. (2014) only used two atoms, but the measured shift was small, 0.2% of the transition linewidth. In this Letter, we demonstrate a large collective Lamb shift for two artificial atoms that significantly exceeds the linewidth and reaches 0.8% of the atomic transition frequency.

Our experimental setup, depicted in Fig. 1, is a superconducting quantum circuit You and Nori (2011); Gu et al. (2017) with two transmon qubits Koch et al. (2007) coupled to a one-dimensional (1D) waveguide. In such superconducting circuits, strong Wallraff et al. (2004); Astafiev et al. (2010); Gu et al. (2017), and even ultrastrong Niemczyk et al. (2010); Forn-Díaz et al. (2017); Kockum et al. (2019); Forn-Díaz et al. (2019), coupling can be engineered between the qubits and photons in the waveguide. Compared to three-dimensional setups, the 1D version strengthens the interaction between qubits and reduces the decay into unwanted modes. These features have enabled many important quantum-optical experiments in 1D waveguide QED in superconducting circuits in the past decade Gu et al. (2017); Roy et al. (2017); Astafiev et al. (2010); Hoi et al. (2011, 2012); van Loo et al. (2013); Hoi et al. (2013, 2015); Forn-Díaz et al. (2017); Liu and Houck (2017); Wen et al. (2018); Mirhosseini et al. (2018); Sundaresan et al. (2019); Mirhosseini et al. (2019) and inspired a wealth of theoretical studies for this platform Gu et al. (2017); Roy et al. (2017); Shen and Fan (2005); Chang et al. (2007); Zhou et al. (2008); Dong et al. (2009); Zheng et al. (2010); Chen et al. (2011); Gonzalez-Tudela et al. (2011); Chang et al. (2012); Koshino and Nakamura (2012); Stannigel et al. (2012); Fan et al. (2013); Lalumière et al. (2013); Peropadre et al. (2013); Tufarelli et al. (2013); Laakso and Pletyukhov (2014); Lindkvist and Johansson (2014); Sathyamoorthy et al. (2014); Shahmoon et al. (2014); Fang and Baranger (2015); Paulisch et al. (2016); Pichler and Zoller (2016); González-Tudela et al. (2017); Guo et al. (2017); Kockum et al. (2018).

As shown in Fig. 1, the transmission line to which the qubits couple is terminated in a capacitive coupling to ground, which is equivalent to placing a mirror in a waveguide. The presence of this mirror separates our experiment from that of Ref. van Loo et al. (2013), where two superconducting qubits were coupled to an open transmission line. In such an open waveguide, the connection between collective decay and the collective Lamb shift entails that the shift always will be smaller than the linewidth Lalumière et al. (2013), and the measurements of elastic scattering in Ref. van Loo et al. (2013) could thus not resolve the collective Lamb shift. Although a splitting in the fluorescence spectrum (inelastic scattering) indicated the presence of the collective Lamb shift, it is not straightforward to extract the size of the shift from the size of the splitting van Loo et al. (2013); Lalumière et al. (2013). In our setup, the presence of the mirror introduces interference effects that suppresses the collective decay more than the collective Lamb shift Sup ; Kockum et al. (2018), allowing us to clearly resolve the shift in simple reflection measurements of elastic scattering. Interestingly, it turns out that these interference effects allow us to couple the two qubits via the transmission line even when one of the qubits is unable to relax into the transmission line.

Device and characterization.

In our device, the interqubit separation $L$ is fixed. However, we can vary the qubit transition frequencies $\omega_{10}$ by applying a local magnetic flux [see Fig. 1(c)] and thus change the effective distance $L/\lambda$, where the wavelength $\lambda=2\pi v/\omega_{10}$ and $v$ is the propagation velocity of the electromagnetic (EM) field in the waveguide Hoi et al. (2015). Since Qubit 2 is placed next to the mirror, it will always be at an antinode of the voltage field in the waveguide [see Fig. 2(b)]. Qubit 1, on the other hand, can be tuned to a voltage node. In this case, Qubit 1 will not couple to the waveguide at its transition frequency, and thus will not contribute to any decay Hoi et al. (2015); Kockum et al. (2018). However, the collective Lamb shift arises due to emission and absorption of virtual photons in all other modes of the continuum in the waveguide, which results in an interaction of strength $\Delta$ between the qubits. This interaction (collective Lamb shift) leads to an avoided level crossing between the two qubits, which shows up as a frequency splitting of $2\Delta$ in reflection measurements of the system using a weak coherent probe at frequency $\omega_{\rm p}$. Our experiment thus clearly demonstrates how the collective Lamb shift has contributions from virtual photons of many frequencies.

We first characterize each of the two transmon qubits through spectroscopy. We detune the transition frequency of one of the qubits far away and measure the amplitude reflection coefficient $\left|r\right|$ of a weak coherent probe tone (i.e., the probe Rabi frequency $\Omega_{\rm p}$ is much smaller than the decoherence rate $\gamma$ of the qubit) as a function of the flux controlling the other qubit’s transition frequency and of the probe frequency $\omega_{\rm p}$. The results are shown in Fig. 2 (Qubit 1 in the left column and Qubit 2 in the right column). For Qubit 1, which is placed at a distance $L$ from the mirror, the spectroscopy data in Fig. 2(c) shows a linewidth narrowing [compare the linecuts A and B from Fig. 2(c), plotted in Fig. 2(e)] and a disappearing response around $4.75\,\mathrm{GHz}$. At this frequency, the effective distance between Qubit 1 and the mirror is $L=7\lambda/4$, which places the qubit at a node for the EM field in the transmission line, as illustrated in Fig. 2(a), and thus effectively decouples the qubit from the transmission line, reducing its relaxation rate to zero Hoi et al. (2015). Qubit 2, on the other hand, is always at an antinode of the EM field in the transmission line [Fig. 2(b)] and thus has an equally strong response at all frequencies [Fig. 2(d), (f)].

We perform further spectroscopy in the full range $4-8\,\mathrm{GHz}$, which is the bandwidth of the cryogenic low-noise amplifier in our experimental setup. The maximum qubit frequency is outside this bandwidth. This data is presented in the supplementary material Sup . From these measurements, we extract Hoi et al. (2013) the qubit relaxation rate $\Gamma$ into the transmission line, the pure dephasing rate $\gamma_{\phi}$ (which also contains contributions from relaxation to other channels), and the speed of light in the transmission line. We further use two-tone spectroscopy, driving at the qubit frequency $\omega_{10}$ and probing around the transition frequency $\omega_{21}$ from the first excited state to the second excited state, to determine the anharmonicity of the qubits. All extracted and derived parameters are summarized in Table 1.

Collective Lamb shift.

We now turn to experiments where both qubits are involved and the collective Lamb shift is measured. We fix the transition frequency of Qubit 2 to $\omega_{10}/2\pi=4.75\,\mathrm{GHz}$, the frequency at which Qubit 1 is at a node of the EM field [see Fig. 2(c)]. We then tune the frequency of Qubit 1 to values around this point and measure the reflection of a weak probe signal on the system for frequencies close to $\omega_{10}$. The results of these measurements are displayed in Fig. 3(a). We observe a clear anti-crossing between the vertical resonance, corresponding to Qubit 2, and the diagonal resonance, corresponding to Qubit 1. The observation of this anti-crossing indicates that the two qubits are coupled on resonance with strength $\Delta$ through a coherent interaction, which must be mediated by the transmission line since the qubits are distant from each other. The minimum size of the separation, shown in the linecut in Fig. 3(c), is $2\Delta\simeq 2\pi\times 38\,\mathrm{MHz}$.

If the qubits were uncoupled, they would have eigenstates $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$, with energies [math], $\hbar\omega_{10}$, $\hbar\omega_{10}$, and $2\hbar\omega_{10}$, respectively. Here, [math] and $1$ denote ground and excited states of a single qubit, respectively; the first number in the kets is for Qubit 1 and the second number is for Qubit 2. Due to the coupling, the eigenstates $|01\rangle$ and $|10\rangle$ are replaced by the symmetric and anti-symmetric eigenstates $|s\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle+|10\rangle\right)$ and $|a\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle-|10\rangle\right)$, respectively, with eigenenergies $\hbar\left(\omega_{10}\pm\Delta\right)$ Lalumière et al. (2013). When the coupling is due to virtual photons, as in our experiment, this thus gives a collective Lamb shift of $2\hbar\Delta$, as illustrated in the inset in Fig. 3(c).

If the two qubits were placed in an open transmission line, it would not be possible to observe the collective Lamb shift in this measurement, since each of the two resonances would have a linewidth set by a relaxation rate $\Gamma=2\Delta$ Lalumière et al. (2013). This is not easily circumvented, since it is the coupling to the transmission line of the two qubits that determines both the relaxation into the transmission line and the strength of the interaction that is mediated via the transmission line. However, the presence of the mirror in our setup breaks this close connection between the linewidth and the collective Lamb shift. In our setup, the collective Lamb shift is given by Kockum et al. (2018); Sup

[TABLE]

where $x_{j}$ denotes the distance of Qubit $j$ from the mirror and $\Gamma_{0}=\sqrt{\Gamma_{1}(\omega_{10})\Gamma_{2}(\omega_{10})}$, with $\Gamma_{j}(\omega)$ the bare relaxation rate of Qubit $j$ at frequency $\omega$ into an open transmission line. This is calculated using the standard master-equation approach with the Born-Markov approximation and tracing out the photonic modes of the transmission line Sup ; Carmichael (1999). When $x_{2}=0$ and $x_{1}$ corresponds to Qubit 1 being at a node of the field in the transmission line, as in Fig. 3, the collective Lamb shift becomes $2\Delta=2\Gamma_{0}$. However, since Qubit 1 is at a node, both the effective relaxation rate of Qubit 1 and the collective decay rate of the two qubits becomes zero. The only contribution from relaxation to the linewidths for the states $|s\rangle$ and $|a\rangle$ is half of $2\Gamma_{2}$, the effective relaxation rate of Qubit 2. In this experiment, we used $\Gamma_{1}\approx\Gamma_{2}$ (giving a shift $2\Delta\approx 2\Gamma_{2}$ and a linewidth $\gamma_{c}\approx\Gamma_{2}$), but we note that the collective Lamb shift could be made many times larger than the linewidths by instead designing the qubits such that $\Gamma_{1}\gg\Gamma_{2}$.

The fact that we can measure the collective Lamb shift even though Qubit 1 ostensibly is decoupled from the transmission line confirms several predictions about how virtual photons influence relaxation and qubit-qubit interaction. The relaxation from Qubit 1 is stimulated by virtual photons in the transmission line at the transition frequency $\omega_{10}$. The relaxation is suppressed when Qubit 1 is placed at a node for the virtual photons at this frequency Hoi et al. (2015). However, Qubit 1 is clearly coupled via virtual photons to Qubit 2. Thus, the virtual photons mediating this coupling, and causing the collective Lamb shift, must have frequencies that are not equal to $\omega_{10}$. In fact, the coupling is given by a sum over all virtual modes at frequencies separate from $\omega_{10}$ Lalumière et al. (2013).

Finally, we note that there are several processes, with real photons, where a strong drive shifts or dresses energy levels of qubits to create an effect that could look similar to what we have observed. To rule out such effects, e.g., the Mollow triplet Mollow (1969) and Autler-Townes splitting Autler and Townes (1955), we measure $\Delta$ as a function of the power $P$ of the coherent probe. The results are shown in Fig. 4. Clearly, the energy shift $\Delta$ is independent of $P$ (before the power is high enough to saturate the qubits), indicating that the collective Lamb shift we measure really is due to virtual photons.

Summary and outlook.

In this Letter, we demonstrated a large collective Lamb shift with two distant superconducting qubits in front of an effective mirror in a 1D transmission-line waveguide. Using interference effects due to the mirror, we overcame previous limitations on the size of the shift compared to the linewidth, allowing us to observe a shift reaching 0.8% of the qubit transition frequency and exceeding the transition linewidth. We explained how future experiments could increase the shift relative to the linewidth even more. This experiment also demonstrated that a qubit can couple to another qubit via the transmission line even though the first qubit is prevented from decaying into the transmission line. These results give further insight into how virtual photons affect both atomic relaxation rates and inter-atomic coupling, and how these effects can be controlled using interference, which could have applications for designing, e.g., devices that process quantum information.

Acknowledgements.

Acknowledgements.

I.-C.H. and J.C.C. would like to thank I. A. Yu and C.-Y. Mou for fruitful discussions. This work was financially supported by the Center for Quantum Technology from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. I.-C.H. acknowledges financial support from the MOST of Taiwan under project 107-2112-M-007-008-MY3. B.S., A.F.K., and P.D. acknowledge support from the Knut and Alice Wallenberg Foundation. G.-D.L acknowledges support from the MOST of Taiwan under Grant No. 105-2112-M-002-015-MY3 and National Taiwan University under Grant No. NTUCC-108L893206. H. I. acknowledges the support by FDCT of Macau under grant 065/2016/A2, by University of Macau under grant MYRG2018-00088-IAPME, and by NNSFC under grant 11404415. J.C.C. acknowledges financial support from the MOST of Taiwan under project 107-2112-M-007-003-MY3. F.N. acknowledges support from the MURI Center for Dynamic Magneto-Optics via the Air Force Office of Scientific Research (AFOSR) award No. FA9550-14-1-0040, the Army Research Office (ARO) under grant No. W911NF-18-1-0358, the Asian Office of Aerospace Research and Development (AOARD) grant No. FA2386-18-1-4045, the Japan Science and Technology Agency (JST) through the Q-LEAP program, the ImPACT program, and CREST Grant No. JPMJCR1676, the Japan Society for the Promotion of Science (JSPS) through the JSPS-RFBR grant No. 17-52-50023 and the JSPS-FWO grant No. VS.059.18N, the RIKEN-AIST Challenge Research Fund, and the John Templeton Foundation.