Entanglement between superconducting qubits and a tardigrade

Quantum and biological systems are seldom discussed together as they seemingly demand opposing conditions. Life is complex,"hot and wet"whereas quantum objects are small, cold and well controlled. Here, we overcome this barrier with a tardigrade -- a microscopic multicellular organism known to tolerate extreme physiochemical conditions via a latent state of life known as cryptobiosis. We observe coupling between the animal in cryptobiosis and a superconducting quantum bit and prepare a highly entangled state between this combined system and another qubit. The tardigrade itself is shown to be entangled with the remaining subsystems. The animal is then observed to return to its active form after 420 hours at sub 10 mK temperatures and pressure of $6\times 10^{-6}$ mbar, setting a new record for the conditions that a complex form of life can survive.

FIG. 1: Sketch of the experiment. a) A tardigrade in the tun state is positioned between the shunt capacitior plates, slightly offset from the Josephson junction, of the transmon qubit B. Qubit A is on the underside and is capacitively coupled to qubit B. The full chip is placed within a 3D copper cavity that is mounted inside a dilution refrigerator and connected to standard microwave electronics for probing. b) Circuit diagram of the two qubits and the tardigrade tun. c) Magnification of the revived tardigrade on transmon qubit. The tardigrade in the tun state is placed in the same position while still attached to a small piece of filter paper during the experiment.
qubit and reintroduced to atmospheric pressure and room temperature. We observe the resumption of its active metabolic state in water. Notably, the tardigrade tun remained at a base temperature below 10 mK and pressures around 6 × 10 −6 mbar for 420 hours (see Supplementary Information (SI) for more details). This is to-date the most extreme exposure to low temperatures and pressures that a tardigrade has been recorded to survive, clearly demonstrating that the state of cryptobiosis ultimately involves a suspension of all metabolic processes [16,17] given that all chemical reactions would be prohibited with all its constituent molecules cooled to their ground states. Figure 1 shows an illustration of the experiment. Two transmon qubits ('qubit A' and 'qubit B') are mounted back to back inside an oxygen-free high thermal-conductivity copper 3D microwave cavity with a TE 101 frequency of 4.521 GHz. The two transmons are positioned about 1 mm apart (twice the thickness of the silicon chips). The two qubits are directly coupled via the shunt capacitor plates on each qubit. We characterize the system using control and readout schemes typically used in superconducting qubits [18]. The frequency of each qubit is tunable, but is always biased close to the maximum in our experiments. These correspond to f A = 3.048 GHz for qubit A and f B = 3.271 GHz for qubit B without tardigrade tun.
For the subsequent cooldown, a single cryptobiotic tardigrade on a 150 µm square piece of cellulose filter paper is placed on the substrate of qubit B, in between the shunt capacitor plates of the transmon qubit, with the tardigrade side closer to the substrate. The filter paper is sufficiently displaced from the transmon plane and interacts very weakly with the transmon electric fields, verified with numerical simulations (see Fig. 2a). The maximum frequency of qubit B is shifted down by 8 MHz to f B = 3.263 GHz, as shown in Fig. 2b, while the maximum frequency of qubit A did not change significantly. The shift can be attributed to the aging of the Josephson junctions and the addition of the tardigrade itself. The addition of the tardigrade results in a change of the effective dielectric medium and contributes to the transmon shunt capacitance of qubit B, consequently shifting its frequency. We simulate the electric fields and capacitance shifts using ANSYS Maxwell where the tardigrade is modelled as a cube of length 100 µm positioned as in the experiment and with dielectric constant varied between 4 to 30, typical values for proteins [19] (the filter paper is also taken into account and verified to have small contribution). Fig. 2b shows that the measured frequency shift corresponds to tardigrade dielectric constant of ε r ≈ 4. Note that this is in the lower range of the dielectric constants for proteins indicating that the contribution from the aging effect is small, in agreement with measurements on qubit A.
This macroscopic behaviour can be understood with a microscopic model where the charges inside the tardigrade are represented as effective harmonic oscillators that couple to the electric field of the qubit via the dipole mechanism. This is a modification of the two-level systems used to model defects in dielectric structures [20]. Due to this coupling, the combined system has different energy gap as compared to just the bare qubit. We model the charges inside the tardigrade as a collection of N harmonic oscillators (with possibly different frequencies) coupled to the qubit. Due to the small observed energy shift it is reasonable to assume that the coupling is weak and could be treated as a perturbation. We therefore split the Hamiltonian and write the following base term where σ z is the Pauli-z operator for the qubit with lower (higher) energy state denoted as |0 (|1 ) and a j (a † j ) is the lowering (raising) operator for the jth oscillator. Note that vanishing energy of the qubit has been set in the middle between its energy levels and the ground state of each oscillator is assigned energy zero. The perturbation term is given by the coupling, where again due to weak interaction we also ignore so-called counter-rotating terms capable of simultaneous excitation of the qubit and an oscillator. Therefore, where for simplicity the same coupling g is used for each oscillator and we write σ + = |1 0| and σ − = |0 1|. One verifies that the ground state of this system is given by |g = |0 . . . 0 for any coupling g, i.e. the qubit and all oscillators are in their respective ground states. The first excited state and its energy can be obtained via the second-order perturbation theory and show the following frequency gap to the ground state: Accordingly, if the sum is negative, the frequency difference is lowered in the presence of interaction. This happens when there are charges in the tardigrade that oscillate faster than the qubit frequency. We shall also need the explicit form of the first excited state which is found to read: where the first subsystem denotes the qubit and the remaining subsystems describe the oscillators. The state |ψ 1 is the superposition of a single excitation in each oscillator, i.e. |ψ 1 = c 1 |10 . . . 0 + · · · + c N |0 . . . 01 with suitable coefficients, and cos θ 2 = 1 − (g 2 /8) j (1/δ 2 j ), where δ j = ω q − ω j is the detuning. Note that this model reduces to the well-known Tavis-Cummings model in the case of identical oscillators [21].
For the second part of the experiment, we couple the qubit B-tardigrade system with qubit A using a pulse sequence shown in Fig. 3. The controlled-NOT gate uses a pulse shape given by Speeding up Waveforms by Inducing Phases to Harmful Transitions (SWIPHT) protocol [22,23]. This pulse sequence ideally produces the entangled state 1 √ 2 (|0 A , e BT + |1 A , g BT ), where we have used subscripts to demarcate the subsystems. To verify the entangled state, we performed quantum state tomography in the four-dimensional subspace of the whole combined tripartite system. We applied 16 different combinations of one-qubit gates on qubit A and dressed states of the joint qubit B-tardigrade system. We then jointly readout the state of both qubits using the cavity [24]. Maximum likelihood estimation [25] was then employed to prevent the resulting density matrix from having nonphysical properties. Comparing the experimental density matrix with the expected density matrix, we find a state fidelity of F = 82 % [26].
In order to understand the entanglement with the tardigrade tun, we first expand the dressed states |g BT and |e BT back into the larger qubit-tardigrade subspace using Eq. (4). Since the states of the oscillators involved in Eq. (4) are orthogonal, the tardigrade is effectively modeled by a qubit. The density matrix of this three-qubit system (qubit A -qubit B -tardigrade) can then be reconstructed from the tomographic data (see SI). Using tangles based on negativity as entanglement quantifiers [27][28][29], we observe that entanglement in various bipartitions of the tripartite system as well as genuine tripartite entanglement are monotonically increasing function of θ, the coupling strength between qubit B and tardigrade tun, see Fig. 3. In particular, for any finite interaction strength the tardigrade is entangled with both qubits.
The results reported here have been measured in a single experimental sequence, i.e. using one tardigrade in its tun state. Two more experiments have been conducted and we wish to point out that it is very important for the revival of the animal to change the external temperature and pressure gently. See SI for their profiles that led to the successful revival.
We conclude by revisiting Bohr's assertion on the impossibility of conducting quantum experiments with living organisms. Our present investigation is perhaps the closest realisation combining biological matter and quantum matter available with present-day technology. While one might expect similar physical results from inanimate object with similar composition to the tardigrade, we emphasise that entanglement is observed with entire organism that retains its biological functionality post experiment. At the same time, the tardigrade survived the most extreme and prolonged conditions it has ever been exposed to, demonstrating that cryptobiosis (latent life) is truly ametabolic. We hope this will stimulate further experiments with the states of the animal being more and more macroscopically distinguishable. Our work provides a first step in the exciting direction of creating hybrid systems consisting of living matter and quantum bits.

TEMPERATURE AND PRESSURE PROFILES
We have attempted three experimental sequences, each one with a different tardigrade. A successful revival was only achieved in the third experimental trial. In this trial a slower venting process was undertaken to implement gentler pressure gradients on the warmup (room pressure was reintroduced in about 1 hour, while typically it takes only about 15 minutes). There are no pressure sensors inside the mixing chamber where the qubit and tardigrade sits. However, the sensors in the outer chambers set an upper limit on the pressures of the mixing chamber. The pressure and temperature profiles in the refrigerator for the experimental run that yielded a successful revival, from 22 Jan 2021 to 25 Jan 2021 (cooldown) and 11 Feb 2021 to 14 Feb 2021 (warmup), are shown in Fig. 2 below. The cooldown took about 2 days to reach sub 10 mK temperatures while the pressure drop to around 6 × 10 −3 mbar took about half an hour. The final pressure is likely to be much lower than 10 −6 mbar [1]. The warmup took about 3 days for temperature and an hour for pressure to return to room values. After the cooldown, the pressure and temperature stabilises to around 6 × 10 −6 mbar and 10 mK, with some fluctuations. A typical 12 hour period is shown in Fig. 3.

RECONSTRUCTION OF THE TRIPARTITE DENSITY MATRIX
The experimentally determined 4 × 4 matrix of the tripartite system composed of qubit A, qubit B and tardigrade tun is not necessarily physical, i.e. positive semi-definite with unit trace, due to experimental errors and noise. Maximum likelihood method was used to reconstruct a physical density matrix ρ closest to the experimental data, by following the procedure described in Ref. [2]. This density matrix is presented in the main text. Note that it is written in the basis {|0g , |0e , |1g , |1e }, where the combined states |g and |e of the qubit B -tardigrade system are used.
In order to isolate the degrees of freedom related to the tardigrade tun we rewrite the density matrix ρ by expanding states |g and |e . To this end we use Eq. (4) of the main text, which we here repeat for convenience: where we introduced qubit notation for the tardigrade degrees of freedom using the fact that states |0 . . . 0 and |ψ 1 are orthogonal. For example, the operator |0g 0e| corresponding to the matrix element of ρ is mapped to the operator in tensor product of three Hilbert spaces: and so on. The expression on the right-hand side is written in the following order H A ⊗ H B ⊗ H T , where the Hilbert spaces describe qubit A, qubit B and effective qubit for the tardigrade tun, respectively. The reconstructed density matrix written in this space explicitly depends on the coupling strength θ are reads: