Photons, quanta and the Fock state: manipulations with a radio-frequency resonator at the quantum level

The world of quantum technology is as rich and tangled as the history of a whole civilization. Some discoveries in this area may surprise us, others lead us into a state of intellectual stupor. And all because the quantum world lives by its own laws, and he often does not care about classical physics. We are used to linking the word "quantum" with calculations that can be made faster and more. However, this is not the only application of quantum technology. Today we look at a study in which quantum mechanics allowed scientists to create an architecture with which you can manipulate a radio frequency resonator at the quantum level. It sounds simple, but in reality the achievement of this was associated with a number of "puzzles". What aspects of quantum science scientists used, how they implemented them, and what exactly this came out of, we learn from the report of the research group. Go.

The basis of the study

The first thing scientists ask themselves is what is the weakest field in quantum mechanics? The answer is one photon. And, it would seem, the detection and manipulation of a single photon should not be a difficult task. However, at megahertz frequencies, this is quite problematic in view of the fact that there are significant thermal fluctuations even at cryogenic temperatures.
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In this study, the scientists used a gigahertz superconducting qubit for direct observation of the quantization of the megahertz radio frequency electromagnetic field. The use of a qubit allows one to gain control over thermal radiation, cooling to the ground quantum mechanical state and stabilization of the Fock * state of a photon.

Fock state * is a state in quantum mechanics when the number of particles is precisely determined.

The problem of thermal "interference" during manipulations with single photons becomes much more noticeable at low frequencies. The accidental appearance and annihilation of photons due to the hot environment causes decoherence * . And this leads to the formation of a combination of random states, from which it is difficult to isolate the quantum state.

Decoherence * is a process of violation of the coordination of oscillatory / wave processes (coherence) due to the interaction of the quantum mechanical system and the environment.

Logically, a similar problem can be solved by applying colder systems to extract the entropy created by the environment. In practice, this solution is referred to as a thermal reservoir.

Scientists have applied the technology of reservoirs in their quantum electrodynamic scheme, which made it possible to effectively cool and manipulate electromagnetic fields at the quantum level.

In their study, scientists were able to gain control over a thermally excited megahertz photon resonator, which made it possible to observe the quantization of radio frequency electromagnetic fields. And manipulation with a quantum state was achieved at the expense of reservoirs. Also, scientists managed to stabilize the single-photon and two-photon states of Fock.

The basis of everything is the reading and control of the resonator by means of the dispersive coupling of the photons of the resonator and the superconducting qubit. However, when there is a gigahertz qubit and a megahertz photon, the connection (connection) between them in the traditional quantum electrodynamic scheme will be extremely weak. But scientists have overcome this obstacle by proposing a new method of connection.

Research results

Image number 1

Through the scheme created by scientists, a very strong connection arises between a qubit and a photon ( 1A ). The scheme consists, among other things, of the following elements:

L _J - Josephson contact, 41 nH (nanogenry);
C _L - capacitor, 11 pF (picofarad);
L - spiral inductor, 28 nH.

At low frequencies, the parasitic capacitance * of the spiral inductor is insignificant, while in the alternative circuit ( 1B ) the frequency of the first transition will be ω _L = 2π x 173 MHz. If, however, there are gigahertz frequencies, CL becomes a short circuit, and the capacitance of the spiral inductor is C _H = 40 fF (femtopharad). In this case, the parallel connection ( 1C ) L _J , L and C _H has the frequency of the first transition in 2π x 5.91 GHz. This configuration of the schemes allows both models to share a Josephson contact.

Parasitic capacitance * is an undesirable capacitive coupling that occurs between the elements of an electronic (in this case, electrodynamic) circuit.

This contact has inductance, which varies depending on the oscillations of the current passing through it. Because of this, the resonant frequency of the high-frequency (HF) mode is shifted in accordance with the number of excitations in the low-frequency (LF) mode and vice versa.

Such cross-Kerr interaction is quantitatively determined by the number of offsets per 1 photon: x = 2√A _H A _L , where the anharmonicity of * HF and LF modes is A _L = h x 495 kHz and A _H = h x 192 MHz.

Anharmonicity * - deviation of the system from the harmonic oscillator.

The cross-Kerr interaction manifests itself as a splitting of the number of photons in the measured microwave reflection S ₁₁ .

As can be seen from the 1D graph, in view of the strong cross-Kerr interaction, quantum oscillations of the Fock photon state (| 0⟩, | 1⟩, | 2⟩ ...) in the resonator lead to a shift in the qubit transition frequency.

The eigenstates of the system were labeled as | j, n⟩, where j = g, e, f, ... is the excitation of the high-frequency mode, and n = 0, 1, 2 ... - the low-frequency mode.

The amplitude of peaks n is proportional to P _n to _ext / to _n , where P _n is the position of the photon number level in the low frequency mode, and to _ext / to _n is the difference between the external connection to _ext / 2π = 1.6 MHz and the width to _n at peak n . In accordance with the Bose-Einstein distribution of the Pn peak heights, the scientists determined the average value of the number of photons n _th = 1.6, which corresponds to a mode temperature of 17 mK (millikelvin).

Bose-Einstein statistics * - the distribution of identical particles with zero or integer spin over energy levels in the state of thermodynamic equilibrium.

The resolution of the peaks of individual photons is determined by the condition for n ≪ x / ħ. Accordingly, the width of the peaks will increase with increasing values ​​of n : _n = (1 + 4 n _th (H)) + 2γ ( n + (1 + 2 n ) n _th ). In this formula, k / 2π = 3.7 MHz is the level of dissipation of the high-frequency mode, and γ / 2π = 23 kHz is the level of dissipation of the low-frequency mode.

In this case, the condition for _n ≪ A _H / ħ makes the transmon (superconducting charge qubit) of the high-frequency mode. This allows you to selectively activate the transitions | g, n ⟩⟷ | e, n and | e, n ⟩⟷ | f, n .

But with a low-frequency mode, everything is different. The line width is only a few MHz, due to the limitation on the side of thermal expansion, which is much larger than A _L. This makes it a kind of harmonic oscillator.

The process of transition of particles between states was carried out through the nonlinearity of the contact by pumping the circuit at ωp frequency. In this process, only 4 photons can interact at a time, when 1 photon in the resonator (low-frequency mode) is annihilated, and already 2 photons are formed on the transmon side.


Image number 2

This method of pumping, in combination with a large difference in the relaxation frequencies of the modes, makes it possible to cool the megahertz resonator to its ground state. The process diagram is shown in 2A .

Cooling will be only if the thermalization rate of the resonator is lower than the transition rate of the excitations from | g, 1⟩ to | g, 0. There is also a second cooling option - through the transition | g, 1⟩⟷ | e, 0⟩. However, this process is two-photon, and therefore requires more swap power.

Figure 2B shows the S ₁₁ measurements (microwave response) at different cooling pump power levels. As we can see from this graph, the best result is achieved when the population level of the ground state is 0.82.

If the population is used as a function of cooperativity * , then it will be seen that with a higher (strong) cooperativity, a sharp decrease in the population index of the ground state will begin. Therefore, the cooling process will not be possible in this situation.

Cooperative * - changes in the state of the system, when the interaction between its elements increases with the course of the process of change in such a way that accelerates this process.

Scientists point out three main factors that limit cooling and lead to what we see on graph 2C - the higher the cooperativity, the worse things are with the population.

The first factor is the thermal population of the qubit. Swapping transfers the population from | g, 1⟩ to | f, 0⟩, however, a reverse process also occurs due to the fact that the level f has a thermal population (albeit very small) - 0.006. This implies the following ratio: P1 / P0 ﹥ Pf / Pg (dashed line at 2C ).

The second factor is that during a strong connection (connection) the pumping hybridizes the states | g, 1⟩ and | f, 0. If g exceeds the decay rate of 2k, then the population of the state | g, 1⟩ starts the transition to | f, 0⟩ and returns back to | g, 1⟩, without having the time to decay to the state | e, 0⟩.


Image # 3: Bypassing the limitations of extra resonance effects of multi-threaded pumping

It is possible to circumvent this limiting factor by “mass character”, that is, several cooling processes | g, n f | f, n -1 simultaneously start. The more such flows, the less swap power is required to achieve the required population of the ground state. Consequently, the effect of extra-resonance effects is reduced.

Moreover, you can combine different processes, | g, n n | f, n -1⟩ and | g, n | f, n + 1, which will allow stabilization of the Fock states of the megahertz resonator.


Image number 4

Finally, the scientists checked the dynamics of the entire system taking into account the reservoirs and thermalization of the megahertz resonator with a time resolution (interval) of 80 ns (nanoseconds). During the measurement of microwave reflection at a certain frequency, the pump was switched on and off for 50 μs (microseconds).

The images above show the results of this test: 4A — dynamics of cooling to the ground state and 4B — stabilization of the Fock single-photon state.

After studying the steady state due to pumping, the latter stopped, which allowed us to observe the process of thermalization of the device.

Scientists summed up their work in several conclusions. First, the system shows good cooling results to the ground state and stabilization of Fock states, but there are certain problems that require further study. First of all, it is an extraresonant effect. This problem can be solved by determining the exact value of A _H and Χ, which will allow to remove non-resonant processes from the frequency range of the cooling process. The second method is to achieve a high population of the ground state before the effect of a strong compound (bond) begins to significantly affect the process. Scientists do not consider the option of reducing the dissipation of a qubit due to the fact that, although this method eliminates the negative effect of non-resonant processes, a strong coupling will occur at a lower pump power.

For more detailed acquaintance with the details of the study I strongly recommend to look into the report of scientists and additional materials to it .

Epilogue

The quantum world, its laws, limitations and advantages are difficult to understand, but it is possible and, most importantly, necessary. One of the most difficult areas in this area is the combination of quantum and classical physicists, that is, the application of quantum technologies to change, control and improve the processes described by classical physics.

In this study, scientists managed to create a quantum device architecture that can manipulate a radio frequency resonator at the quantum level. The researchers themselves are optimistic about the future of their offspring. According to them, this can give impetus to create a similar, but much more complex and large-scale system that can help in the study of bodies in Bose-Hubbard systems. Scientists also point out that their creation can serve as a link between quantum technologies and physical systems in the megahertz frequency range. This device can also be used to improve NMR (nuclear magnetic resonance) and even in radio astronomy.

Thank you for your attention, stay curious and have a great work week guys.

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