Continuous wave single photon transistor based on a superconducting circuit

We propose a microwave frequency single photon transistor which can operate under continuous wave probing, and represents an efficient single microwave photon detector. It can be realized using an impedance matched system of a three level artificial ladder-type atom coupled to two microwave cavities connected to input/output waveguides. Using a classical drive on the upper transition, we find parameter space where a single photon control pulse incident on one of cavities can be fully absorbed into hybridized excited states. This subsequently leads to series of quantum jumps in the upper manifold and the appearance of a photon flux leaving the second cavity through a separate input/output port. The proposal does not require time variation of the probe signals, thus corresponding to a passive version of single photon transistor. The resulting device is robust to qubit dephasing processes, possesses low dark count rate for large anharmonicity, and can be readily implemented using current technology.

Electronic transistors-devices where a weak electrical signal controls a strong probe from a source-lie at the heart of modern electronics, and has led to vast development of classical computing devices. By analogy, in the realm of quantum computing a similar device was contrived, where a single photon control pulse triggers the transmission of a strong coherent probe, and was named a single-photon transistor (SPT) [1]. The operation of the SPT device proposed in Ref. [1] is based on a time-dependent control of the drive and a strong atom-photon interaction. In the optical domain single-photon transistors along these lines were realized with neutral atoms embedded in an optical cavity [2][3][4], a quantum dot in a waveguide [5,6], or an ultracold gas with Rydberg interactions [7,8]. So far the achieved efficiencies have been limited, but if this is improved, a single-photon transistor could represent a powerful tool for coherent state manipulation and quantum information processing [9]. Importantly, a SPT can also serve as an efficient single-photon detector (SPD), as it amplifies a single photon signal by a large gain.
Recently, a microwave frequency range counterpart of quantum optics-circuit quantum electrodynamics (cQED) [10]-has emerged as a highly promising platform for quantum computation [11][12][13][14]. Based on high-quality superconducting microwave cavities combined with Josephson junction-based artificial atoms, it enables a strong light-matter coupling even at the single photon level, and allows studying numerous nonlinear microwave quantum optics phenomena [15][16][17][18]. The development of a simple and efficient single microwave photon detector is still an open question [19]. The suggested realizations include SPDs based on current-biased Josephson junctions [20], catching an inverted time-controlled pulse [21,22], transmon chain linked with nonreciprocal elements [23], and double quantum dot structures [24]. Also, several schemes for cQED-based single-photon transistors have been proposed [25,26]. Ultimately, however, these proposals rely on active time control of the system and input single photon pulse, which complicates the detection process and limits the applicability. Lately such a time dependent protocol based on impedance-matched artificial Λ atoms was proposed [27] and experimentally realized [28]. This protocol can be extended to perform time-independent detection [29] if highly anharmonic systems with long coherence time can be constructed.
Here, we propose a single-photon transistor which can operate under continuous wave (cw) probe conditions, where a single photon control pulse triggers an avalanche of gain photons. It represents a passive device which does not require signal and probe timing, largely extending its applicability. The proposed device is robust to imperfections and is particularly insensitive to dephasing. The generic idea relies on a three level ladder atom [ Fig. 1(b)], with its lower transition weakly coupled to the first input cavity, and the upper transition strongly driven by a classical source as well as strongly coupled to a second output cavity. A single photon entering the input port transfers the atom to the excited subspace through an impedance matching mechanism similar to Refs. [27][28][29][30][31][32]. In the excited subspace, a number of quantum jumps between dressed atom-cavity states leads to an enhanced output signal.
System and Hamiltonian.-As a particular realization of cw microwave SPT we propose a superconducting artificial atom with three states jgi, jei, and jfi (qutrit), which is coupled to two separate microwave cavities (modesâ 1 and a 2 ), both connected to input-output waveguide channels with coupling constants κ 1 and κ 2 [see sketch in Fig. 1(a)]. The qutrits should have versatile connectivity [13,33,34] and sizeable anharmonicity, making flux [35] and fluxonium [36] types desirable. Its lower jgi-jei transition is resonantly coupled to a cavity modeâ 1 with a perturbative coupling g 1 , while the upper jei-jfi transition is strongly coupled to cavityâ 2 with strength g 2 [ Fig. 1 Here ω eg , ω fe , ω cav;1 , ω cav;2 denote energy separations between qutrit levels and energies of microwave cavities, sequentially (ℏ ¼ 1). ω s is the single input photon central frequency and ω d is the frequency of the classical drive. σ þ mn ¼ jmihnj (σ − mn ¼ jnihmj) denotes qutrit raising (lowering) operator, and σ mm ¼ jmihmj.Ĥ wgd;j describes the coupling to the waveguides j ¼ 1, 2 [37a]. The expression forĤ sys assumes infinitely large anharmonicity, which precludes parasitic couplings to other-than-resonant qutrit transitions, and we assume long decay and dephasing times for the qutrit. These assumptions will be revisited later. Also, in the following we consider zero detuning for the incoming single photon pulse, δ e ¼ δ cav;1 ¼ 0, a resonant microwave drive, δ f ¼ 0, and a cavity resonant to the upper transition, δ cav;2 ¼ 0.
Operational principle.-First, a single photon pulse enters through the input channel, which is loaded by the joint qubit-cavity system with Hilbert space fjgi; jei; jfig ⊗ jn 1 i ⊗ jn 2 i ≡ jm; n 1 ; n 2 i (m ¼ g, e, f).
In the input stage we consider a weak excitation such that n 1 is restricted to vacuum or a single excitation. Then, the relevant subspace of states contains the ground state jg; 0; 0i, the first cavity excited state jg; 1; 0i, and the subspace fje; 0; n 2 i; jf; 0; n 2 ig which we call excited states. Because of the strong couplings g 2 and Ω, the excited states become hybridized, and it is convenient to introduce dressed metastable states. For the lowest cavity 2 occupation with n 2 ¼ 0, 1 this embeds a subspace M ¼ fj1i; j2i; j3i; j4ig ( Fig. 1(c) and [37b]), and higher states can be included analogously. Because of the admixture of the je; 0; 0i level, each dressed state is coupled to jg; 1; 0i with a modified constant g 1m . Considering g 1 to be perturbative, the coupling of the first cavity to the dressed states M works as an effective decay channel. By controlling the drive strength Ω and coupling parameters, this effective total decay rate Γ set can be made equal to the coupling of the cavity to the first waveguide κ 1 , reducing the system to an impedance matched Λ system attached to a single-sided waveguide [37c]. Such impedance matching (IM) has already proven to be useful for cQED circuits, leading to photo detection [28], as well as proposals for microwave downconversion [30,31] and gates for flying qubits [32]. This allows for a full absorption of a single photon pulse by the metastable states, leading to near-unity single photon switching and photon detection without the need for temporally varying control fields.
Once the setting stage to the excited states manifold is completed, the second decay channel κ 2 leads to quantum jumps between the dressed states [ Fig. 1(a), red dashed lines], where a series of jumps within the full fje; 0; n 2 i; jf; 0; n 2 ig (n 2 ¼ 0; 1; …; N 2 ) subspace occur. Radiative jumps between the states lead to a flux leaving cavity 2 through the output port, resulting in an enhancement of the signal. Once the system recovers to the ground state jg; 0; 0i, the SPT duty cycle is finalized. The SPT can thus serve as a highly efficient single-photon detector, and corresponds to a microwave version of a single photon avalanche diode.
Input: impedance matching.-First, we characterize the single photon input stage. To define the IM condition analytically we exploit the effective operator theory [38] to estimate the decay rate Γ set from jg; 1; 0i to the metastable states [37d]. Considering g 1 < g 2 , Ω and up to N 2 ¼ 2 photons, we find which provides a good estimate for κ 2 =g 2 ≳ 1.5, where the relevant processes happen within the lowest Fock states. The setting rate for smaller κ 2 =g 2 can be derived by increasing To test the IM condition numerically, we exploit the input-output theory for the Hamiltonian (1) by deriving the Heisenberg equations of motion for the system operators, and assuming a weak coherent input. The efficiency of the setting stage is quantified by numerically calculating the reflection coefficient jr 1 j 2 ¼ jhâ out;1 i=hâ in;1 ij 2 at the input port. In the simulation a reflection minimum is obtained at the impedance matched setting rate κ 1 ¼ Γ set [ Fig. 2(a)]. The dotted curve corresponds to the full Heisenberg equation calculation, and the solid curve shows the analytical solution for a reflection coefficient of a waveguide coupled to a Λ system [37c,39], with Γ set provided by Eq. (3). The dependence of the setting rate on the classical microwave drive strength Ω allows fine tuning the IM condition. In Fig. 2(b) we show the dependence of Γ set =g 2 on Ω, fixing κ 1 ¼ 0.005g 2 and g 1 ¼ 0.05g 2 . Plotting the setting rate for three different values of κ 2 we can find a value of Ω=g 2 for which IM holds. Output: Gain of SPT.-To describe the gain of the transistor, we exploit the Heisenberg equations of motion derived using the input-output relations for two coupling channels combined with a Gaussian-shaped single photon pulse [37e]. For temporal widths of the pulse being larger than τ > κ −1 1 and IM arranged, an incoming single photon excitation is fully transferred to the excited states of the qutrit which are highly mixed with the mode of cavity 2. Averaging the Heisenberg equation with the wave function corresponding to the single photon input jΨ in i, we can extract the intensity of outgoing photons in the second waveguide, hâ † out;2âout;2 i ¼ κ 2 hâ † 2â2 i ≡ I out;2 . In Fig. 3(a) we plot the temporal dependence of input 1 and output 2 photon numbers. For a long Gaussian pulse containing a single photon [N in;1 ¼ R dtI in;1 ðtÞ ¼ 1], we can on average get N out;2 ¼ R dtI out;2 photons at the output of the second cavity before the system recovers to the ground state. This represents the gain of the single-photon transistor, which describes the effective amplification of the single photon signal. The full counting statistics can be obtained by changing to the wave-function Monte Carlo approach. This confirms that a single input photon leads to numerous emitted photons [37f].
To investigate the gain we consider the input stage to be completed, setting jΨ start i ¼ je; 0; 0i, and calculate the occupation of cavity 2 using a density matrix approach, truncating the cavity Fock space at N 2 ¼ 10 excitations. The gain is highly sensitive to the system parameters: cavity 1 to qutrit coupling g 1 , microwave drive strength Ω, and output coupling κ 2 . Additionally, these parameters set the optimal (IM) value for the input coupling κ 1 , and thereby the bandwidth of the single-photon detector. In Fig. 3(b) we show the gain and bandwidth as a function of the coupling g 1 for varying Ω=g 2 . The plot shows a fast increase of the bandwidth with the coupling constant g 1 , as it sets the rate at which single photons can get to the excited subspace. Reciprocally, this corresponds to a growth of the  140503-3 recovery rate and a reduction of the gain. Thus, there is a trade-off between gain and bandwidth, indicating that a medium g 1 =g 2 ratio is favored for optimal detection.
The gain and bandwidth dependence as a function of κ 2 is shown in Fig. 3(c). Choosing g 1 =g 2 ¼ 0.05 and Ω=g 2 ¼ 2 in order to get high input bandwidth, we find that both gain and bandwidth are nonmonotonic functions of κ 2 . In particular, we are interested in the region of 0.5 < κ 2 =g 2 < 3, where the gain increases rapidly, while κ 1 is relatively large.
Imperfections.-We now turn to imperfections of the scheme. First, we plot the SPT gain as a function of qubit decay (solid line) and pure dephasing (dashed line) rate γ [37g] for three values of κ 2 =g 2 [see Fig. 4(a)]. While strong decay naturally tends to decrease the gain by introducing an extra nonradiative recovery channel, pure dephasing does not influence the gain, thus removing the need for long qubit coherence time, as opposed to e.g. Refs. [26,27]. We also find that even for larger qubit decay the IM conditions can be always satisfied.
Second, we account for a finite qutrit anharmonicity A and introduce the residual qutrit-cavity couplings defined by the Hamiltonian [37h] and the associated detunings defined by the rotated energy frame ofĤ A ¼ Aσ ee þ Aσ ff þ Aâ † 1â1 . Here, the extra terms which couple the ground and the excited states in the absence of a signal photon lead to a nonzero dark count rate of the detector.
There are two separate dark count rates induced by the driving of the lower transition in the system as contained in Eq. (2). The first involves a cycle jg; 0; 0i − je; 0; 0i − jg; 0; 1i − jg; 0; 0i with emission of a single photon at a rate Γ ðsÞ dark . Importantly, this process does not include radiative transitions within the excited manifold. Thus it is not amplified by SPT, and can be discriminated for large SPT gains. The second process corresponds to ground-to-excited state transitions jg; 0; 0i − je; 0; 0i − jg; 0; 1i − je; 0; 1i − je; 0; 0i happening at a rate Γ ðmÞ dark , which projects the system to the excited manifold, launches a photon avalanche, and needs to be strongly suppressed.
To access the rates directly we find the full counting statistics using the wave function Monte Carlo approach and study the no-jump evolution of the system [37h], allowing us to estimate the dark count rates for the single and enhanced processes. The results are shown in Fig. 4(b). Additionally, the steady state dark count rates can be calculated using an effective operator approach [37h], valid for large anharmonicities, which gives Γ ðsÞ dark;ss ≈  [30]. Decreasing g 2 to 2π × 120 MHz, setting κ 2 ¼ g 2 , g 1 ¼ 2π × 6 MHz, and Ω ¼ 2π × 240 MHz gives a gain of 172 photons with 2π × 0.6 MHz bandwidth, enhanced dark count rate of 2π × 660 Hz, and single dark count rate 2π × 14.4 kHz.
Finally, we note that for using the device as a singlephoton detector the gain photons need to be measured. For the above scenario with a gain of approximately 200 we estimate that the signal is distributed on ∼90 modes. With a heterodyne detection setup this output field can be measured with an efficiency of 95% with only a 0.02 dark count probability and better performance can be achieved at higher gain [37i]. Alternatively the expected signal begins to be within range of calorimetric detection schemes [40].
Conclusion.-We have presented a scheme for a singlephoton transistor based on the impedance-matched superconducting circuit, which operates in the cw regime and allows for an on-demand single microwave photon detection. The scheme can realistically lead to an output of several hundred photons, tolerates high pure dephasing rates, and keeps low dark count rates.