# Entanglement between superconducting qubits and a tardigrade

K. S. Lee  School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore  Y. P. Tan  School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore  L. H. Nguyen  School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore  R. P. Budoyo  Centre for Quantum Technologies, National University
of Singapore, Singapore  K. H. Park  Centre for Quantum Technologies, National
University of Singapore, Singapore  C. Hufnagel  Centre for Quantum
Technologies, National University of Singapore, Singapore  Y. S. Yap  Faculty
of Science and Centre for Sustainable Nanomaterials (CSNano), Universiti
Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia  Centre for Quantum
Technologies, National University of Singapore, Singapore  N. Møbjerg
Department of Biology, University of Copenhagen, Denmark  V. Vedral
Department of Physics, University of Oxford, United Kingdom  Centre for
Quantum Technologies, National University of Singapore, Singapore  Department
of Physics, National University of Singapore, Singapore  T. Paterek  Institute
of Theoretical Physics and Astrophysics, University of Gdańsk, Poland  R.
Dumke  School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore  Centre for Quantum Technologies, National University of
Singapore, Singapore

###### Abstract

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.

In his highly influential 1933 lectures titled “Light and Life” BOHR1933
Niels Bohr espoused an analogy between physicists’ attempt to characterize the
atom and biologists’ attempt to characterize the cell. Living cells are made
of ordinary matter, and therefore fully subject to chemical analysis, but the
matter is organized in a complex and intricate way. Bohr proposed that in
order to study the chemistry of the cell, the organization had to be
destroyed. On the other hand, to study the organization, one has to operate at
a level at which the chemistry is invisible. Bohr therefore proposed that the
chemical basis of an organism and its organizational hierarchy are
complementary properties, and could never be studied simultaneously.

Here we not only go beyond Bohr’s complementarity by entangling a living
system to two superconducting quantum bits, but we also expose an additional
twist in areas where quantum physics, chemistry and biology all overlap.
Namely, the living system we study is a tardigrade which is capable of
suspending its living functions for an indefinite amount of time if the
external conditions are deemed to be adverse enough for survival moberg2021 .
In that sense, and this is a point that Bohr certainly missed, it is possible
to do a quantum and hence a chemical study of a system, without destroying its
ability to function biologically.

Our experiments could be classified within the field of quantum biology
Lambert2012  which studies the possibility of quantum effects in biologically-
relevant processes such as photosynthesis Cao2020 , olfaction Brookes2017  or
animal navigation Wiltschko2019 , but is also concerned with quantum features
of entire organisms. Along this direction, a systematic body of work with
matter-wave interferometers revealed wave properties of molecules with growing
complexity and holds promise for similar outcomes measured with viruses
Arndt1999 ; Hackermller2003 ; Gerlich2011 . Living photosynthetic bacteria
have been observed to strongly couple to light Coles2017  and under certain
assumptions the results obtained could be interpreted in terms of entanglement
between the quantised light and light-sensitive parts of bacteria Marletto2018
. Our experiments involve a multi-cellular eukaryote that is a much more
complex unit of life interacting electrically with quantum bits and for which
conclusions about quantum features are unavoidable.

In our experiments, we use specimens of a Danish population of Ramazzottius
varieornatus Bertolani and Kinchin, 1993 (Eutardigrada, Ramazzottiidae). The
species belongs to phylum Tardigrada comprising of microscopic invertebrate
animals with an adult length of 50-1200 $\mu$m moberg2018_2 . Importantly,
many tardigrades show extraordinary survival capabilities Mobjerg11  and
selected species have previously been exposed to extremely low temperatures of
50 mK Paul1950  and low Earth orbit pressures of $10^{-19}$ mbar Jnsson2008 .
Their survival in these extreme conditions is possible thanks to a latent
state of life known as cryptobiosis moberg2021 ; Mobjerg11 . Cryptobiosis can
be induced by various extreme physicochemical conditions, including freezing
and desiccation. Specifically, during desiccation, tardigrades reduce volume
and contract into an ametabolic state, known as a “tun”. Revival is achieved
by reintroducing the tardigrade into liquid water at atmospheric pressure. In
the current experiments, we used dessicated R. varieornatus tuns with a length
of 100-150 $\mu$m. Active adult specimens have a length of 200-450 $\mu$m. The
revival process typically takes several minutes.

We place a tardigrade tun on a superconducting transmon qubit and observe
coupling between the qubit and the tardigrade tun via a shift in the resonance
frequency of the new qubit-tardigrade system. This joint qubit-tardigrade
system is then entangled with a second superconducting qubit. We reconstruct
the density matrix of this coupled system experimentally via quantum state
tomography. Finally, the tardigrade is removed from the superconducting 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 $6\times
10^{-6}$ 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 Keilin ; clegg2001  given that all chemical
reactions would be prohibited with all its constituent molecules cooled to
their ground states.

Figure 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.

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 TE101 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 Lin2021 . 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 $\mu$m
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
$f^{\prime}_{B}=3.263$ 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 $\mu$m positioned as in the experiment and with dielectric
constant varied between 4 to 30, typical values for proteins li2013  (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 $\varepsilon_{r}\approx 4$. 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 sarabi2016 . Due to this coupling, the
combined system has different energy gap as compared to just the bare qubit.

Figure 2:  a) Numerical simulations of the electric field along the surface of
the tardigrade for an exemplary potential difference of 20 mV across the shunt
capacitor plates (shown in orange). The tardigrade in the tun state was
simulated as a cube of length 100 $\mu$m, with dielectric constant of $4$,
attached to a 150 $\mu$m square filter paper with dielectric constant of 4.5.
As seen, the filter paper is sufficiently displaced from the transmon plane
and does not interact significantly with the electric fields. b) Frequency
shift expected from tardigrade, simulated with typical permittivity values for
proteins (in the range 4 to 30). The observed frequency shift of -8 MHz
corresponds to a dielectric constant of about 4 for the tardigrade tun.

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

$\displaystyle H_{0}$
$\displaystyle=-\frac{1}{2}\hbar\omega_{q}\,\sigma_{z}+\hbar\sum_{j=1}^{N}\omega_{j}\,a^{\dagger}_{j}a_{j},$
(1)

where $\sigma_{z}$ is the Pauli-$z$ operator for the qubit with lower (higher)
energy state denoted as $|0\rangle$ ($|1\rangle$) and $a_{j}$
($a_{j}^{\dagger}$) is the lowering (raising) operator for the $j$th
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,

$\displaystyle V$ $\displaystyle=\frac{\hbar
g}{2}\sum_{j=1}^{N}(a^{\dagger}_{j}\sigma_{-}+a_{j}\sigma_{+}),$ (2)

where for simplicity the same coupling $g$ is used for each oscillator and we
write $\sigma_{+}=|1\rangle\langle 0|$ and $\sigma_{-}=|0\rangle\langle 1|$.
One verifies that the ground state of this system is given by
$|g\rangle=|0\dots 0\rangle$ 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:

$\Delta
f=\frac{1}{2\pi}\left(\omega_{q}+\frac{g^{2}}{4}\sum_{j=1}^{N}\frac{1}{\omega_{q}-\omega_{j}}\right).$
(3)

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:

$\left|e\right\rangle=\cos\frac{\theta}{2}\,\left|1\right\rangle\left|0\dots
0\right\rangle+\sin\frac{\theta}{2}\,\left|0\right\rangle\left|\psi_{1}\right\rangle,$
(4)

where the first subsystem denotes the qubit and the remaining subsystems
describe the oscillators. The state $\left|\psi_{1}\right\rangle$ is the
superposition of a single excitation in each oscillator, i.e.
$\left|\psi_{1}\right\rangle=c_{1}\left|10\dots
0\right\rangle+\dots+c_{N}\left|0\dots 01\right\rangle$ with suitable
coefficients, and
$\cos\frac{\theta}{2}=1-(g^{2}/8)\sum_{j}(1/\delta_{j}^{2})$, where
$\delta_{j}=\omega_{q}-\omega_{j}$ is the detuning. Note that this model
reduces to the well-known Tavis-Cummings model in the case of identical
oscillators Tavis1968 .

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 Economou2015 ; Premaratne2019 . This
pulse sequence ideally produces the entangled state
$\frac{1}{\sqrt{2}}(\left|0_{\mathrm{A}},e_{\mathrm{BT}}\right\rangle+\left|1_{\mathrm{A}},g_{\mathrm{BT}}\right\rangle)$,
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 Filipp2009 . Maximum likelihood estimation James2001  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 $F=82$ % Jozsa1994 .

In order to understand the entanglement with the tardigrade tun, we first
expand the dressed states $\left|g_{\mathrm{BT}}\right\rangle$ and
$\left|e_{\mathrm{BT}}\right\rangle$ 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 CKW ; Zhang2017 ; Ou2007 , we
observe that entanglement in various bipartitions of the tripartite system as
well as genuine tripartite entanglement are monotonically increasing function
of $\theta$, 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.

Figure 3: Entanglement with the tardigrade. a) Tomography of the combined
tripartite system in the four-dimensional subspace spanned by the states of
qubit A and dressed states of qubit B and tardigrade tun together. The
fidelity of the density matrix reconstructed via maximum likelihood estimation
(green) to the ideal state (blue) is $82\%$. b) The Hadamard gate (H) followed
by the CNOT gate was used to produce the state presented in panel a). c)
Quantum entanglement in the microscopic model as a function of $\theta$,
interaction strength between the tardigrade tun and qubit B. The $\pi$-tangle
quantifies genuine tripartite entanglement, i.e. a form of quantum
correlations that cannot be reduced to pairwise entanglement. See SI for
formal definitions.

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.

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## Additional Information

This work was supported by the National Research Foundation and the Ministry
of Education of Singapore, and the Polish National Agency for Academic
Exchange NAWA Project No. PPN/PPO/2018/1/00007/U/00001. The authors thank Dr.
Farshad Foroughi (CNRS Grenoble) for the assistance in the fabrication of the
transmon qubits used in the experiment.

Author Contrubutions All authors researched and wrote this paper.

Supplementary Information Supplementary information is available for this
paper.

Competing Interests The authors declare that they have no competing financial
interests.

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