Catching and counting photons

Title: Number-Resolved Photocounter for Propagating Microwave Mode

Authors: Rémy Dassonneville, Réouven Assouly, Théau Peronnin, Pierre Rouchon, Benjamin Huard

Institution: Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France

Manuscript: Published in Physical Review Applied [1], Open Access on arXiv

Introduction

Quantum technologies, based on superconducting circuits and microwave photons, are rapidly developing. At the heart of these devices are superconducting qubits, highly customizable and capable of strong interactions with microwave photons [2]. This basic building block enables everything from multi-qubit quantum processors to state of the art sensors. One capability missing from the toolkit of superconducting qubits is the ability to detect propagating photons, which can be used to send quantum information over long distances. Developing a method to resolve the number of photons contained in a wavepacket traveling down a transmission line could unlock the ability to construct quantum networks, entangle remote qubits, and build quantum sensors.

The strong interactions between qubits and photons make qubits an ideal device for photon detection. In a stationary mode, the photons can be held for a long time, allowing a qubit to distinguish between 0, 1, 2, … photons [3]. However, a propagating wavepacket travels so quickly that the qubit only has enough time to determine if there are an even or odd number of photons. In this work, the authors devise a scheme to catch an arbitrary signal propagating down a microwave transmission line and efficiently count the number of photons in the wavepacket [1]. I first describe the device and its components used to make this possible. Next, I go through the catch protocol, useful for holding a wavepacket for long enough to be measured. I break down the photon counting measurement that resolves $N$ photons in only $\log_2{N}$ measurements. Finally, I discuss the potential applications of the device and protocol developed in this work.

Device

Figure 1: (Top) A wavepacket traveling down the transmission line is converted into a stationary buffer mode. The packet is swapped into a long lived memory mode to be measured by a qubit and readout system. (Bottom) False color image of the physical implementation of the device using superconducting circuits. The buffer mode is in orange, JRM in the purple inset, memory in blue, qubit in green, and readout in gray. Figure adapted from [1].

The device consists a series of carefully designed microwave components as shown in Figure 1. A propagating wavepacket first encounters a buffer cavity, a superconducting $LC$ circuit that can hold photons with frequency matched to the resonance of the cavity $\omega = 1/\sqrt{LC}$ . The buffer cavity is strongly coupled to the transmission line in order to capture the traveling wavepacket and temporarily hold it, making it a stationary mode. Resolving the number of photons in the signal requires enough time to make the measurements, however the strong coupling between the transmission line and buffer cavity, used to capture the wavepacket, results in the signal quickly leaking out of the stationary mode and back into the propagating line. To ensure there is enough time to measure the photon number, the state is swapped into a long lived memory cavity that can store the signal while it is being read out. This is done using a Josephson ring modulator (JRM), which swaps the states of the buffer and memory cavities. Once the state has been transferred into the memory, a qubit and readout system can be used to determine the number of photons present. After measuring the state, the JRM swaps the state back into the buffer where is it quickly emitted into the transmission line. This resets the system and allows for the next operation to proceed.

Catching Photons

A Josephson ring modulator (JRM) is a device used to swap the states of two cavity modes. In this work, it is used to transfer the wavepacket captured in the leaky buffer cavity into the long lived memory cavity. The JRM is pumped at the frequency corresponding to the difference between the buffer (resonant frequency $\omega_b/2\pi = 10.220$ GHz) and memory ( $\omega_m/2\pi = 3.745$ GHz) cavities frequencies. This provides the energy required to transfer a state from the buffer to the memory cavity. The pump enables a beam splitter interaction between the buffer and memory, described by the Hamiltonian shown in Equation 1.

$\mathcal{H}_{bs} = g p(t) b m^{\dagger} + h.c.$ (Equation 1)

$g$ is the strength of the pump, $p(t)$ is the pulse shape of the pump, $b$ is the annihilation operator of the buffer mode, and $m^{\dagger}$ is the creation operator for the memory mode. When the pump is on, the state present in the buffer mode is swapped into the memory at a rate $g$ . For example, if we start with 1 photon in the buffer cavity and 0 photons in the memory, after a time $t=\pi/g$ , the buffer will contain 0 photons and the memory will have 1 photon. The authors use this interaction to move the wavepacket from the buffer cavity to the memory for measurement. This process can also be used in reverse; a photon contained in the memory cavity can be transferred into the buffer by applying the same pump. The authors use this interaction to reset the device by emptying out the memory cavity. The pump is turned on to swap the memory state into the buffer, where the wavepacket quickly escapes into the transmission line.

Counting Photons

Once the wavepacket is successfully transferred into the long lived memory cavity, the qubit and readout are used to count the number of photons present. The authors make a series of measurements to resolve the photon number of the wavepacket. Each measurement involves allowing the qubit and memory cavity states to interact for a carefully chosen amount of time. At the end of each measurement the qubit state (ground, g or excited, e) is read out and recorded. The authors devise protocol that requires making the minimal number of measurements: resolving up to $N$ photon with only $\log_2{N}$ qubit measurements. The measurements are designed such that the series of recorded qubit states represent a binary decomposition of the photon number. The binary decomposition is a way to represent any integer as a sum of powers of 2. An integer is represented as a series of bits (which take the value 0 or 1), where the $k^{\mathrm{th}}$ bit determines if $2^k$ is included in that sum. For example, $13 = 1(2^0) + 0(2^1) + 1(2^2) + 1(2^3)$ , so for 13, bit 0 = 1, bit 1 = 1, bit 2 = 1, and bit 3 = 1. Here, the qubit state after each measurement (either g or e) represents the value of the bit being measured (which can be either 0 or 1). Each photon number is then identified as a unique series of g’s and e’s (or 0’s and 1’s). In this work, the authors measure wavepackets that contain up to 3 photons, using two measurements to distinguish between 0, 1, 2, and 3 photons as shown in Table 1.

Bit 0	Bit 1	Number
0	0	0 = 0(2⁰) + 0(2¹)
1	0	1 = 1(2⁰) + 0(2¹)
0	1	2 = 0(2⁰) + 1(2¹)
1	1	3 = 1(2⁰) + 1(2¹)

Table 1: The series of qubit states resulting from the measurements represent a unique way to identify the photon number in the memory cavity. In this work, the authors are able to resolve up to 3 photons with a series of two measurements.

Binary decomposition measurement protocol

The measurement harnesses the interaction between the qubit and memory cavity a described by the Hamiltonian in Equation 2. The memory creation and annihilation operators are represented as $m^{\dagger}$ and $m$ . The qubit ground and excited states are $\left|g\right\rangle$ and $\left|e\right\rangle$ .

$\mathcal{H}_{qm} = -\chi m^{\dagger}m \left|e\right\rangle \left\langle e \right|$ (Equation 2)

Without the interaction between the qubit and memory cavity, the Hamiltonian of the just the qubit would look like $\mathcal{H}_{q} = \omega_q \left|e\right\rangle \left\langle e \right|$ , where $\omega_q$ is the transition frequency of the qubit. When we add in the interaction the full Hamiltonian can be expressed as $\mathcal{H}_{\mathrm{total}} = \mathcal{H}_{q} + \mathcal{H}_{qm} = (\omega_q - \chi m^{\dagger}m )\left|e\right\rangle \left\langle e \right|$ . The combination $m^{\dagger}m$ is the operator version of the number of photons, n, in the memory cavity. By comparing the total Hamiltonian with the one of just the qubit, we see that the effect of the interaction is to modify the transition frequency of the qubit (represented by everything before the $\left|e\right\rangle \left\langle e \right|$ ). So now, the qubit transition frequency is dependent on the number of photons in the memory cavity ( $n = m^{\dagger}m$ ). For every additional photon in the memory cavity, the qubit transition frequency shifts by $\chi$ .

To access the memory cavity photon number requires multiple similar, but subtly slightly different measurements, all relying on the interaction that causes a qubit frequency shift proportional to the number of memory photons. In each measurement, the authors entangle the cavity state with that of the qubit. This is done by placing the qubit in a superposition state $\frac{1}{\sqrt{2}} (\left|g\right\rangle + \left|e\right\rangle)$ using a $\pi/2$ rotation about the x-axis and allowing it to interact with the memory cavity state, according to Equation 2, for a carefully chosen time. During this interaction time, $\tau$ , the qubit state acquires a phase that is proportional to the number of photons, $n$ , in the cavity at a rate of $\chi$ . The total phase acquired is $\phi = n \chi \tau$ . The qubit is then projected back onto the z-axis using a second $\pi/2$ rotation.

In the first measurement, the goal is to distinguish between even and odd photon numbers, 0/2 or 1/3, the 0^th bit of information. The interaction time is chosen to be $\tau_0 = \frac{2 \pi}{2 \chi}$ so that when the photon number is even the phase acquired is an even multiple of $\pi$ and when the photon number is odd the phase acquired is an odd multiple of $\pi$ . The second $\pi/2$ rotation is performed around the -x-axis, rotated by $\pi$ relative to the original axis. If there are an even number of photons in the memory (0 or 2), the second $\pi/2$ rotation just undoes the first one since the qubit superposition phase is $\phi = l\pi$ with $l =0, 2$ . Since the qubit state remains $\frac{1}{\sqrt{2}} (\left|g\right\rangle + e^{il\pi} \left|e\right\rangle) = \frac{1}{\sqrt{2}} (\left|g\right\rangle + \left|e\right\rangle)$ after the interaction, the qubit ends up back in the ground state. If there are an odd number of photons in the memory (1 or 3), the qubit phase is $\phi = m\pi$ , $m=1, 3$ . The qubit state is $\frac{1}{\sqrt{2}} (\left|g\right\rangle + e^{im\pi} \left|e\right\rangle) = \frac{1}{\sqrt{2}} (\left|g\right\rangle - \left|e\right\rangle)$ . The second $\pi/2$ rotation acts in concert with the first, combining to form a $\pi$ pulse, which takes the qubit to its excited state.

In the second measurement, the interaction time is halved to be $\tau_1 = \frac{1}{2} \tau_0 = \frac{2 \pi}{4 \chi}$ to distinguish between 0 and 2 photons (or 1 and 3 photons), the 1^st bit of information. The axis of the second $\pi/2$ rotation is conditioned upon the result on the first measurement. If the first measurement results in the qubit remaining in the ground state (photon number is even), the second pulse is performed around the -x-axis, rotated by $\pi$ relative to the original axis. If there are 0 photons in the memory, the qubit returns to the ground state and if there are 2 photons, the qubit is excited. If the first measurement results in the qubit being excited (photon number is odd), the second pulse is performed around the -y-axis, rotated by $3\pi/2$ relative to the original axis. If there is 1 memory photon, the qubit ends in the ground state and if there are 3 photons, the qubit is excited.

The series of qubit states depending on the memory photon number is shown in Table 2. This protocol realizes the binary decomposition shown in Table 1 where the qubit ground state (g) serves as a 0 and the excited state (e) as 1.

1st Result	2nd Result	Photon Number
g (0)	g (0)	0
e (1)	g (0)	1
g (0)	e (1)	2
e (1)	e (1)	3

Table 2: Binary decomposition protocol results in a series of qubit state readouts. This set of measurements results corresponds to the decomposition of the wavepakcet photon number into its constituent bits.

Counting more photons

In order to measure wavepackets with even larger photon number, the series of measurements can be extended to extract more bits of information. For each bit, a measurement similar to the ones described above would be performed, where the axis of rotation for the second $\pi/2$ pulse depends on the result of previous measurement.

Large integer values can be represented by only a few bits, for example integers up to $1024$ can be represented by only $\log_2(1024) = 10$ bits. Since it takes only one measurement per bit, large numbers of photons up to $N$ can be resolved in only $\log_2 N$ measurements. This provides a way to efficiently measure lots of information encoded in quantum states with many photons.

Future outlook

The authors devise a protocol to catch an arbitrary wavepacket by capturing it in a buffer cavity and swapping it into a long lived memory cavity. Once the wavepacket is in the memory cavity, a qubit is used to count the number of photons contained using the minimal number of measurements. In this work, the authors are able to distinguish between 0, 1, 2, 3 photons in a wavepackets.

This device can serve as a central component of a quantum network. Quantum information can be encoded into a wavepacket with different superpositions of photon numbers. The information can be transported along a transmission line to a secondary location, where the device presented in this work can capture and readout out the information stored by assessing the photon number of the wavepacket. The photon detection and counting component can also be used to entangle remote qubits. An emitter qubit can be coupled to a transmission line such that the qubit state is encoded as a wavepacket with a superposition of different photon numbers. This wavepacket can be transported along a transmission line to a remote receiver qubit. By counting the wavepacket photon number, the state of the receiver qubit can be conditioned on the number of photons in the wavepacket, and by extension the state of the original emitter qubit. Finally, by combining the two techniques described in this work, the device can be used as a quantum sensor, with potential applications in dark matter searches and gravitational wave detection. An arbitrary microwave signal can be caught efficiently and distinguished from backgrounds by measuring the number of photons in the wavepacket.

References

[1] Dassonneville, R., Assouly, R., Peronnin, T., Rouchon, P. & Huard, B. Number-resolved photocounter for propagating microwave mode. Physical Review Applied 14 044022 (2020).

[2] Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).

[3] Schuster, D. I. et al. Resolving photon number states in a superconducting circuit. Nature 445, 515–518 (2007).

Akash Dixit builds superconducting qubits and couples them to 3D cavities to develop novel quantum architectures and search for dark matter.

Thanks to Sapphire Lally for thoughtful and insightful edits.

Catching and counting photons

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