A team of physicists has successfully created a unique quantum state of matter called a Bose-Einstein condensate (BEC) out of molecules. The team is led by Sebastian Will at Columbia University, who received a U.S. National Science Foundation Faculty Early Career Development award to support his research. The findings were published in the journal Nature.

While BECs have previously been achieved with atoms of a single element, this is the first creation of a molecular BEC. Its greater period of stability will allow scientists to test longstanding theories in quantum phenomena, including superconductivity, superfluidity and more. Molecular BECs also hold the potential to outperform single-element BECs by creating longer-ranging interactions in quantum simulators, allowing for more complicated models.

Credit: Sebastian Will/ Will Lab/ Columbia University

Single-element BECs have expanded the understanding of concepts such as the wave nature of matter and led to the development of technologies such as quantum gas microscopes and quantum simulators.

Researchers plan to use molecular BECs to explore more quantum phenomena, including new types of superfluidity, a state of matter that flows without experiencing any friction. They also hope to use their molecular BEC-based quantum simulator to help guide the development of new quantum materials.

Credit: Sebastian Will/ Will Lab/ Columbia University

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Physicists create first-ever Bose-Einstein condensate made of molecules - National Science Foundation (.gov)

When you hear the term quantum mechanics, you no doubt think about some weird and counterintuitive stuff things like electrons being both particles and waves or cats being simultaneously dead and alive. However, despite the almost-unbelievable subatomic spookiness, our modern societies would not be possible if quantum mechanics were not true.

In recognition of a century since the theory was developed, the United Nations has declared 2025 to be the International Year of Quantum Science and Technology, a year for both celebrating past successes and planning for a future in which people across the globe can benefit from these crucial technologies.

In the late 1800s, physicists had a pretty good handle on the rules governing nature. They had worked out the laws of motion, electromagnetism, and heat. This mastery of physics set into motion the cultural change we call the Industrial Revolution. Yet, a few niggling mysteries remained. For example, the color of light emitted from a glowing steel foundry wasnt as purple as expected.

From this and other observations, physicists realized that the world of the atom acted very differently than the one we experience with our senses. The first baby steps toward understanding this unfamiliar world were taken in 1900, when German physicist Max Planck postulated that the energy carried by light and other electromagnetic waves was related to the waves frequency.

The following decades led to many discoveries and refinements that slowly exposed an atomic world governed by a confusing mlange of seemingly contradictory behaviors. Electrons and light sometimes acted like a tiny baseball and other times like the undulating surface of a pool crowded with children. It was a confusing time.

The situation became greatly clarified in 1925 when Austrian physicist and bon-vivant Erwin Schrdinger wrote down what is now called the Schrdinger equation. This equation successfully described the spectrum of light emitted by atoms, marking the moment when the atom transitioned from a chemical curiosity to an object whose mysterious properties could be harnessed for the betterment of humanity.

Over the past century, our understanding of the quantum realm has grown exponentially, and atomic physics ushered in the age of electronics. Transistors and other semiconductor technologies made our modern world possible. Our command of the atom led to the LEDs that power the screens of our laptops and smartphones. Even the humble laser point, with which we bedevil our cats, is a handheld form of a quantum-driven technology that enables global telecommunications.

And the evolution of quantum technology is far from over. Researchers continue to develop new materials with the potential to create room-temperature superconductors, which would revolutionize how we transport power around the globe. Improvements in the design of solar cells will help us efficiently extract energy from the sun and allow us to combat the dangers inherent in our fossil fuel-based economy.

The understanding of quantum mechanics is also leading to a transformation of the computer industry. Quantum computers, which dont rely on the binary on or off of traditional computers, can solve complex problems that take modern computers years to crack in a fraction of a second. For instance, the cryptographic algorithms that secure credit card transactions and private communications between nation-states are easily broken by quantum computers.

The declaration by the UN of the International Year of Quantum Science and Technology both plays homage to the role of quantum theory in developing our modern technological society and looks forward to future innovations. It also will inspire national physics organizations around the world to develop education and outreach programs to teach citizens the impact that this transformational physics has had on humanity.

Over the next year, readers can expect to learn a great deal more about how quantum mechanics has and will continue to impact society.

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Why the U.N. declared 2025 the International Year of Quantum Science and Technology - Big Think

image:

Physicists at UC Berkeley immobilized small clusters of cesium atoms (pink blobs) in a vertical vacuum chamber, then split each atom into a quantum state in which half of the atom was closer to a tungsten weight (shiny cylinder) than the other half (split spheres below the tungsten). By measuring the phase difference between the two halves of the atomic wave function, they were able to calculate the difference in the gravitational attraction between the two parts of the atom, which matched what is expected from Newtonian gravity.

Credit: Cristian Panda/UC Berkeley

Dark energy a mysterious force pushing the universe apart at an ever-increasing rate was discovered 26 years ago, and ever since, scientists have been searching for a new and exotic particle causing the expansion.

Pushing the boundaries of this search, University of California, Berkeley physicists have now built the most precise experiment yet to look for minor deviations from the accepted theory of gravity that could be evidence for such a particle, which theorists have dubbed a chameleon or symmetron.

The experiment, which combines an atom interferometer for precise gravity measurements with an optical lattice to hold the atoms in place, allowed the researchers to immobilize free-falling atoms for seconds instead of milliseconds to look for gravitational effects, besting the current most precise measurement by a factor of five.

Though the researchers found no deviation from what is predicted by the theory spelled out by Isaac Newton 400 years ago, expected improvements in the precision of the experiment could eventually turn up evidence that supports or disproves theories of a hypothetical fifth force mediated by chameleons or symmetrons.

The ability of the lattice atom interferometer to hold atoms for up to 70 seconds and potentially 10 times longer also opens up the possibility of probing gravity at the quantum level, said Holger Mller, UC Berkeley professor of physics. While physicists have well-tested theories describing the quantum nature of three of the four forces of nature electromagnetism and the strong and weak forces the quantum nature of gravity has never been demonstrated.

"Most theorists probably agree that gravity is quantum. But nobody has ever seen an experimental signature of that," Mller said. "It's very hard to even know whether gravity is quantum, but if we could hold our atoms 20 or 30 times longer than anyone else, because our sensitivity increases with the second or fourth power of the hold time, we could have a 400 to 800,000 times better chance of finding experimental proof that gravity is indeed quantum mechanical."

Aside from precision measurements of gravity, other applications of the lattice atom interferometer include quantum sensing.

"Atom interferometry is particularly sensitive to gravity or inertial effects. You can build gyroscopes and accelerometers," said UC Berkeley postdoctoral fellow Cristian Panda, who is first author of a paper about the gravity measurements set to be published this week in the journal Nature and is co-authored by Mller. "But this gives a new direction in atom interferometry, where quantum sensing of gravity, acceleration and rotation could be done with atoms held in optical lattices in a compact package that is resilient to environmental imperfections or noise."

Because the optical lattice holds atoms rigidly in place, the lattice atom interferometer could even operate at sea, where sensitive gravity measurements are employed to map the geology of the ocean floor.

Dark energy was discovered in 1998 by two teams of scientists: a group of physicists based at Lawrence Berkeley National Laboratory, led by Saul Perlmutter, now a UC Berkeley professor of physics, and a group of astronomers that included UC Berkeley postdoctoral fellow Adam Riess. The two shared the 2011 Nobel Prize in Physics for the discovery.

The realization that the universe was expanding more rapidly than it should came from tracking distant supernovas and using them to measure cosmic distances. Despite much speculation by theorists about what's actually pushing space apart, dark energy remains an enigma a large enigma, since about 70% of the entire matter and energy of the universe is in the form of dark energy.

One theory is that dark energy is merely the vacuum energy of space. Another is that it is an energy field called quintessence, which varies over time and space.

Another proposal is that dark energy is a fifth force much weaker than gravity and mediated by a particle that exerts a repulsive force that varies with the density of surrounding matter. In the emptiness of space, it would exert a repulsive force over long distances, able to push space apart. In a laboratory on Earth, with matter all around to shield it, the particle would have an extremely small reach.

This particle has been dubbed a chameleon, as if it's hiding in plain sight.

In 2015, Mller adapted an atom interferometer to search for evidence of chameleons using cesium atoms launched into a vacuum chamber, which mimics the emptiness of space. During the 10 to 20 milliseconds it took the atoms to rise and fall above a heavy aluminum sphere, he and his team detected no deviation from what would be expected from the normal gravitational attraction of the sphere and Earth.

The key to using free-falling atoms to test gravity is the ability to excite each atom into a quantum superposition of two states, each with a slightly different momentum that carries them different distances from a heavy tungsten weight hanging overhead. The higher momentum, higher elevation state experiences more gravitational attraction to the tungsten, changing its phase. When the atom's wave function collapses, the phase difference between the two parts of the matter wave reveals the difference in gravitational attraction between them.

"Atom interferometry is the art and science of using the quantum properties of a particle, that is, the fact that it's both a particle and a wave. We split the wave up so that the particle is taking two paths at the same time and then interfere them at the end," Mller said. "The waves can either be in phase and add up, or the waves can be out of phase and cancel each other out. The trick is that whether they are in phase or out of phase depends very sensitively on some quantities that you might want to measure, such as acceleration, gravity, rotation or fundamental constants."

In 2019, Mller and his colleagues added an optical lattice to keep the atoms close to the tungsten weight for a much longer time an astounding 20 seconds to increase the effect of gravity on the phase. The optical lattice employs two crossed laser beams that create a lattice-like array of stable places for atoms to congregate, levitating in the vacuum. But was 20 seconds the limit, he wondered?

During the height of the COVID-19 pandemic, Panda worked tirelessly to extend the hold time, systematically fixing a list of 40 possible roadblocks until establishing that the wiggling tilt of the laser beam, caused by vibrations, was a major limitation. By stabilizing the beam within a resonant chamber and tweaking the temperature to be a bit colder in this case less than a millionth of a Kelvin above absolute zero, or a billion times colder than room temperature he was able to extend the hold time to 70 seconds.

He and Mller published those results in the June 11, 2024, issue of Nature Physics.

In the newly reported gravity experiment, Panda and Mller traded a shorter time, 2 seconds, for a greater separation of the wave packets to several microns, or several thousandths of a millimeter. There are about 10,000 cesium atoms in the vacuum chamber for each experiment too sparsely distributed to interact with one another dispersed by the optical lattice into clouds of about 10 atoms each.

"Gravity is trying to push them down with a force a billion times stronger than their attraction to the tungsten mass, but you have the restoring force from the optical lattice that's holding them, kind of like a shelf," Panda said. "We then take each atom and split it into two wave packets, so now it's in a superposition of two heights. And then we take each one of those two wave packets and load them in a separate lattice site, a separate shelf, so it looks like a cupboard. When we turn off the lattice, the wave packets recombine, and all the quantum information that was acquired during the hold can be read out."

Panda plans to build his own lattice atom interferometer at the University of Arizona, where he was just appointed an assistant professor of physics. He hopes to use it to, among other things, more precisely measure the gravitational constant that links the force of gravity with mass.

Meanwhile, Mller and his team are building from scratch a new lattice atom interferometer with better vibration control and a lower temperature. The new device could produce results that are 100 times better than the current experiment, sensitive enough to detect the quantum properties of gravity. The planned experiment to detect gravitational entanglement, if successful, would be akin to the first demonstration of quantum entanglement of photons performed at UC Berkeley in 1972 by the late Stuart Freedman and former postdoctoral fellow John Clauser. Clauser shared the 2022 Nobel Prize in Physics for that work.

Other co-authors of the gravity paper are graduate student Matthew Tao and former undergraduate student Miguel Ceja of UC Berkeley, Justin Khoury of the University of Pennsylvania in Philadelphia and Guglielmo Tino of the University of Florence in Italy. The work is supported by the National Science Foundation (1708160, 2208029), Office of Naval Research (N00014-20-1-2656) and Jet Propulsion Laboratory (1659506, 1669913).

Experimental study

Measuring gravitational attraction with a lattice atom interferometer

26-Jun-2024

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Precision instrument bolsters efforts to find elusive dark energy - EurekAlert

Quantum circuit model

Let us consider the multimode circuit QED system in Fig.1a, namely a Josephson junction coupled to a lumped-element transmission line. As detailed in the Methods, the quantum Hamiltonian has been derived from the circuit Lagrangian depending on C and , respectively, the capacitance and inductance matrices. The system degrees of freedom are described by the vector , containing independent flux variables41, indicated in Fig.1a. To get the final form, we have performed a simultaneous diagonalization of C and via the basis change =P. Through a Legendre transform and identifying as the junction phase =kP0kk, we have obtained the classical Hamiltonian (for an analogous procedure see42). Finally, by quantizing the degrees of freedom, we get the quantum Hamiltonian in the charge gauge

$$hat{{{{{{{{mathcal{H}}}}}}}}}={hat{{{{{{{{mathcal{H}}}}}}}}}}_{J}+sumlimits_{k=1}^{{N}_{m}}hslash {omega }_{k}{hat{a}}_{k}^{{{{dagger}}} }{hat{a}}_{k}+ihat{N}sumlimits_{k=1}^{{N}_{m}}{g}_{k}({hat{a}}_{k}^{{{{dagger}}} }-{hat{a}}_{k}),$$

(1)

where ({hat{{{{{{{{mathcal{H}}}}}}}}}}_{J}=4{E}_{C}{hat{N}}^{2}-{E}_{J}cos hat{varphi }) and ({hat{a}}_{k}^{{{{dagger}}} }) is the bosonic creation operator for the kth mode and ({g}_{k}=2sqrt{hslash {omega }_{k}{E}_{C}}{P}_{0k}) its coupling to the junction. Here we used the following sum rule, due to the normalization identity PTCP=I,

$${P}_{00}^{2}+frac{1}{4{E}_{C}}sumlimits_{k=1}^{{N}_{m}}frac{{g}_{k}^{2}}{hslash {omega }_{k}}=1.$$

(2)

a Representation of a circuit consisting of a Josephson junction (red) of Josephson energy EJ and capacitive energy EC coupled to a transmission line resonator. The fluxes used to describe the circuit are indicated with black dots. b Energy bands of the uncoupled junction in terms of the quasi-charge for EJ=0.5EC. c Frequencies of the transmission line modes versus the mode number. d Junction-line couplings in the charge gauge as a function of the mode frequency for different values of the impedance Z. For these last two panels: plasma frequency p=5EC and free spectral range =0.2EC (see definition in the text).

The frequencies and couplings in (1) depend on the capacitive and inductive matrices. Illustrative values are shown in Fig.1c, d. A lumped-element circuit possesses an upper frequency cutoff, known as the plasma frequency p. Overall, the line has three key parameters: the impedance (Z=sqrt{L/C}), the plasma frequency ({omega }_{p}=2/sqrt{LC}), and the free spectral range ({{Delta }}=frac{pi }{sqrt{LC}}frac{1}{{N}_{m}}), where Nm is the number of modes. The phase transition is expected to emerge in the thermodynamic limit Nm, that is equivalent to 0.

It is insightful to consider the basis diagonalizing the last two terms of (1), namely (leftvert N,{{{{{{{bf{n}}}}}}}}rightrangle={leftvert Nrightrangle bigotimes }_{k=1}^{{N}_{m}}leftvert {alpha }_{N,k},{n}_{k}rightrangle), where (leftvert {alpha }_{N,k},{n}_{k}rightrangle={D}_{k}({alpha }_{N,k})leftvert {n}_{k}rightrangle), with ({D}_{k}(alpha )=exp (alpha {hat{a}}_{k}^{{{{dagger}}} }-{alpha }^{*}{hat{a}}_{k})) the displacement operator43 for the kth mode with ({alpha }_{N,k}=iNfrac{{g}_{k}}{hslash {omega }_{k}}), and n is a vector with the number of photons in each mode. The only nontrivial term in (1) is the junction Hamiltonian: the dressing of the junction by the line can be determined by inspecting its matrix elements in such a basis. As detailed in the Methods, the coupling to the line effectively produces a renormalized Josephson energy

$${widetilde{E}}_{J}=exp left(-frac{1}{2}sumlimits_{k=1}^{{N}_{m}}frac{{g}_{k}^{2}}{hslash {omega }_{k}^{2}}right){E}_{J},$$

(3)

independent of the number of photons in the modes. The remaining effect is a renormalized capacitive energy

$${widetilde{E}}_{C}={P}_{00}^{2}{E}_{C} , < ,{E}_{C}.$$

(4)

P00 vanishes in the thermodynamic limit and has been taken to be exactly zero in a recent study of the phase transition23. However, the way it approaches zero in the thermodynamic limit is important and cannot be neglected. Indeed, also the renormalized Josephson energy ({widetilde{E}}_{J}) tends to zero for 0.

The properties of the infinite system will hence be determined by the asymptotic behavior of their ratio. We computed such analytical expression by injecting the numerically evaluated values of gk, k and P00 for different values of the impedance Z and decreasing values of . The results are summarized in Fig.2. One can see that both ({widetilde{E}}_{C}) and ({widetilde{E}}_{J}) decrease for all values of Rq/Z while is decreased (Fig.2b, c). However, while ({widetilde{E}}_{C}) monotonically decreases while increasing Rq/Z, ({widetilde{E}}_{J}) first decreases and then increases. The ratio of the two quantities ({widetilde{E}}_{J}/{widetilde{E}}_{C}) instead displays a striking behavior (Fig.2d): for small enough it sharply decreases from 1 to very small values and then grows again. Remarkably, the curves for different values of all cross for Z=Rq, with the curves for larger sizes (smaller ) tending to smaller values for Rq/Z<1, and to larger values for Rq/Z>1. The inset Fig.2e shows that the slope of the curves at Z=Rq grows logarithmically with 1. We can conclude that in the thermodynamic limit the ratio ({widetilde{E}}_{J}/{widetilde{E}}_{C}) tends to 0 for Rq/Z<1 (apart from Rq/Z=0, for which no renormalization can occur) and to infinity for Rq/Z>1.

a In the thermodynamic limit the full system can be understood as a renormalized junction. b, c Renormalized capacitive and Josephson energies computed from (4) and (3) for different values of . d Ratio of the renormalized Josephson and charging energies. e Slope of the ratio at Z=Rq showing logarithmic growth in the thermodynamic limit. For these plots EJ=EC and p=2EC.

This points at the two expected phases: for Rq/Z<1 the charging energy dominates (so-called insulating behavior), for Rq/Z>1 the charging energy is negligible (so-called superconducting behavior). We emphasize that our theoretical analysis predicts a singular point for Z=Rq, independently of the bare ratio EJ/EC, and that the compact or extended nature of the junction phase does not play a role here.

To complete our investigation and fully characterize the emergence of the two phases we have applied exact diagonalization techniques44,45 to our Hamiltonian (1) for finite-size systems. To extend the reachable sizes, we have taken full advantage of Josephson potential periodicity and of Bloch theorem. Indeed, we can introduce the eigenstates of the junction Hamiltonian (leftvert nu,srightrangle={e}^{inu varphi }{u}_{nu }^{s}(varphi )), where is the quasi-charge and s is the band index. In our charge gauge representation, the full Hamiltonian is block diagonal and for each we need to diagonalize the following matrix:

$${hat{H}}_{nu }= sumlimits_{s}{varepsilon }_{nu }^{s}leftvert nu,srightrangle leftlangle nu,srightvert+sumlimits_{k=1}^{{N}_{m}}hslash {omega }_{k}{hat{a}}_{k}^{{{{dagger}}} }{hat{a}}_{k}\ +isumlimits_{k=1}^{{N}_{m}}{g}_{k}({hat{a}}_{k}^{{{{dagger}}} }-{hat{a}}_{k})sumlimits_{s,r}leftlangle nu,rrightvert hat{N}leftvert nu,srightrangle leftvert nu,rrightrangle leftlangle nu,srightvert,$$

(5)

where ({varepsilon }_{nu }^{s}) are the eigenenergies of the bare junction with [0.5,0.5] (Brillouin zone). Note that the spectrum is even with respect to , so we can restrict to positive values only.

Illustrative energy bands for the bare junction are reported in Fig.1b. An example of the obtained energy bands for the interacting system is shown in Fig.3 versus Z. Different colors correspond to different values of the free spectral range for the same p (i.e., to different numbers of modes), and the energies are rescaled by . For Rq/Z0, the junction and the modes are decoupled: the spectrum is given by the bare junction bands with replicas due to a finite number of bosons in the modes. For finite Rq/Z instead, the interaction manifests in energy anticrossings. To investigate a quantum phase transition, the behavior of the ground state and of the low-lying excited states is crucial. While increasing Rq/Z, the first energy band becomes narrower, corresponding to a reduced effective charging energy (see Fig.2b). However, the curvature of the first band cannot change, since the sum rule (2) implies that ({widetilde{E}}_{C}ge 0). This means that the ground state is always at =0. The first excited band instead changes its shape while varying Z. When the system is not coupled (Rq/Z=0), it is simply a one-photon replica of the first band. Instead, when Rq/Z is increased, the slope changes (see Fig.3f, g). The low-energy spectrum of the full system at low impedances is reminiscent of the bare junction spectrum, although at much smaller energies. This qualitative behavior of the first few bands occurs for different sizes, although at smaller energies for smaller (different colors in Fig.3). Interestingly, the rescaled bands overlap in the best way for Z=Rq, confirming the universality observed in Fig.2d. Away from this point, for both larger and smaller values of Z, the rescaled spectra do not overlap any longer. This is reminiscent of the finite-size scaling of continuous phase transitions.

ag Bands for different impedances Z, as indicated in the subplot titles. The different line colors correspond to different free spectral ranges (see legend). The differences of the energies and the ground state energy EG are plotted and the vertical axis is rescaled by . In these plots EJ=0.5EC and p=2EC.

In the exact diagonalization studies, we considered an extended phase and applied Blochs theorem. Now, since the full Hamiltonian for our finite-size system does not couple different quasi-charges , the results for a compact phase are independent of the phase being fundamentally extended or compact. Indeed, we found that the ground state always occurs for =0, which corresponds to a periodic wavefunction. This independence of the phase transition on the extended/compact nature of the junction phase was also discussed in22. For more general shunts (e.g., inductive) an extended phase is relevant27 and this may be needed to describe transport through the junction.

In Fig.4a we plot the ground state energy versus Rq/Z: surprisingly, it has a weak dependence on the system size, and no precursor of a critical behavior is apparent. Figure4b instead reports the energies of the first few excited states, separated from the ground state by an energy of the order of for all the values of Z. In the thermodynamic limit, the system is expected to be gapless for all values of Z: hence, the phase transition needs to emerge with a different mechanism. Indeed, around Z=Rq the first three excited levels exhibit an anticrossing. This corresponds to the observation we already made for Fig.3 that the first excited band passes from being a one-photon replica to a dressed Josephson band. For a compact phase, this occurs around Z=Rq, with the splitting of the resulting anticrossing (but not the position) depending on EJ, as displayed in Fig.5a. As in Fig.3, while decreasing all the rescaled spectra exhibit a universal behavior at Z=Rq. Hence, when approaching the thermodynamic limit, for a fixed EJ the anticrossing becomes narrower and narrower, meaning that the signature of the phase transition first emerges in the first excited states. At the same time, the spectrum is gapless in the 0 limit so that a singular behavior in the first excited state necessarily reflects on the ground state.

a Ground state energy for different free spectral ranges. b Corresponding dependence of the first few excited state energies for a compact phase (=0). c Junction charge fluctuations in the ground state for different free spectral ranges. d Energy difference between the first two bands at the edge of the Brillouin zone =0.5. In all the panels EJ=0.5EC and p=2EC.

a Dependence on EJ of the anticrossing involving the first three excited states for a compact phase (=0). Here =0.55EC and p=2EC. b Energy difference between the first two bands at the edge of the Brillouin zone (=0.5) showing the universality at Rq/Z=1 for a wide range of EJ. Solid lines =0.55EC, dashed lines =0.41.

While a compact phase is the only relevant quantity for the ground state of the system, the response of the system to external perturbations can depend on the full bands. To better highlight the behavior of the low-lying bands, in Fig.4d we show the behavior of the gap between the first and second bands at the edge of the Brillouin zone. The ratio of the gap and the free spectral range displays a behavior analogous to the renormalized parameters of Fig.2, with the gap closing for larger systems for Rq/Z<1. This means that the low energy spectrum is approaching a capacitive free particle" behavior. The opposite is true for Rq/Z>1. As shown in Fig.5b, this behavior extends to larger values of EJ.

Importantly, however, the ground state does not behave in the same way, as can be seen for example by computing observables for the junction degrees of freedom. As an example, we show in Fig.4c the charge fluctuations ({sigma }_{N}^{2}={{{{{{{rm{tr}}}}}}}}({hat{rho }}_{J}{hat N}^{2})-{{{{{{{rm{tr}}}}}}}}{({hat{rho }}_{J}hat N)}^{2}), with ({hat{rho }}_{J}) the reduced density matrix for the junction (analogous results are obtained, for example, for ({{{{{{{rm{tr}}}}}}}}({hat{rho }}_{J}cos hat{varphi }))). For Rq/Z>1 the charge fluctuations have a strong dependence on the system size. For Rq/Z<1 instead, the size dependence is much smaller, and the fluctuations never fall below the Cooper pair box value that is obtained for Rq/Z0 (horizontal dashed line). This, however, does not mean that the transition does not affect the ground state, as signaled by the different size dependence of the charge fluctuations on the two sides of Rq/Z=1. The emergence of the singularity in the ground state in the thermodynamic limit is, however, slow, and we can only see a precursor.

We believe that this peculiar difference of behavior between the ground state and the excited states is at the origin of much of the recent controversy over this phase transition, specifically over the characterization of the so-called insulating state. In particular, in the thermodynamic limit, the ground state is not approaching a purely capacitive behavior. However, the response of the system to a gate charge (i.e., to a change in the quasi-charge) changes sharply at the transition point.

To understand the fate of the higher frequency modes at the transition point, we focus on the compact-phase Hamiltonian, given by equation (5) for =0. For EJ=0, the dressed excited bands (charge states) energies are ({varepsilon }_{0}^{n}({E}_{J}=0)=4{widetilde{E}}_{C}{n}^{2}), with n an integer. In this limit, the effect of the transition on the photons can be understood by simply plotting the Z-dependent mode energies and the same energies shifted by the first dressed charge state ({varepsilon }_{0}^{1}({E}_{J}=0)). We show these two sets of energy branches in Fig.6. The two lowest solid and dashed curves are respectively the first photon energy 1 and ({varepsilon }_{0}^{1}({E}_{J}=0)). Note that they cross at Z=Rq. With a finite EJ the crossing is replaced by the anticrossing highlighted in Fig.5a. Above these levels, for energies small enough with respect to p, also the kth photonic mode energy k crosses the energy ({varepsilon }_{0}^{1}({E}_{J}=0)+hslash {omega }_{k-1}) at Z=Rq. This degeneracy is again lifted for EJ0. Hence, around the transition point, all the single photon states will hybridize. Note that the crossing of the first two curves,which is essentially determined by the universality highlighted in Fig.2, is robust with respect to the intrinsic ultraviolet cutoff given by the Josephson plasma frequency p. For the excited states, there is eventually an important shift of the crossing point at high energies, that would be absent in the limit p (a non-dispersive line). The same shift is also present for much smaller free spectral ranges, of the order of the ones in the experiment39, as shown by the red line in Fig.6. For modes above the third, also states involving higher dressed bands play a significant role. Moreover, states with more than one photon are also present. For energies small enough with respect to p (equidistant modes) these are also degenerate with the single-photon energies at the transition point, increasing the number of levels that participate the hybridization.

Single photon energies (solid lines) and the same energies shifted by the energy of the first excited renormalized junction band for =0 (dashed lines) with EJ0, =0.5EC and p=5EC. The nth mode energy and the nth level of the shifted branch have the same color. Their intersection point is highlighted by a black dot. The solid red line shows the intersections for a much smaller free spectral range =0.02EC.

Recent experiments have observed a striking signature of the transition from the dispersion of high-frequency modes39. With the results of our exact diagonalization, we can calculate the linear-response photonic spectral function

$$D(E)=sumlimits_{n}frac{{gamma }^{2}}{{gamma }^{2}+{(E-{E}_{n}+{E}_{G})}^{2}}| leftlangle Grightvert sumlimits_{k}({a}_{k}+{a}_{k}^{{{{dagger}}} })leftvert {E}_{n}rightrangle {| }^{2},$$

(6)

where En ((leftvert {E}_{n}rightrangle)) is the nth excited eigenenergy (eigenstate) of the full system, EG ((leftvert Grightrangle)) the ground energy (state) and is a phenomenological broadening. Here, as in the experiment, we have considered a SQUID that is equivalent to a junction with EJ that can be tuned by a magnetic field. Illustrative spectra versus the external magnetic flux are shown in Fig.7. We observe a clear change of sign of the photon frequency dispersion across the transition, as observed in39. This effect was linked to a capacitive/inductive behavior of the junction and here we see that it microscopically originates from the crossings observed in Fig.6. The broadening observed in39 for high energy modes is instead not present because of the moderate size of the system simulated here with exact diagonalization techniques. In fact, it relies on multiple resonances between single and multi-photon states37. In Fig.7, one can see the state belonging to the shifted set of levels acquiring a finite single-photon component near the crossing point for /00.5, that is for ({E}_{J} neq 0). For larger and smaller impedances, this state is dark, i.e., it does not contribute to this spectral function.

Color plot of D(E) around the bare energy of a photonic mode versus normalized flux bias (0=h/2e). The panels correspond to different values of Rq/Z with EJ=0.1EC, =0.5EC and p=5EC.

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Emergent quantum phase transition of a Josephson junction coupled to a high-impedance multimode resonator - Nature.com