Non-degenerate, three-wave mixing with the Josephson ring modulator
The Josephson ring modulator (JRM) is a device, based on Josephson tunnel junctions, capable of performing non-degenerate mixing in the microwave regime without losses. The generic scattering matrix of the device is calculated by solving coupled quantum Langevin equations. Its form shows that the device can achieve quantum-limited noise performance both as an amplifier and a mixer. Fundamental limitations on simultaneous optimization of performance metrics like gain, bandwidth and dynamic range (including the effect of pump depletion) are discussed. We also present three possible integrations of the JRM as the active medium in a different electromagnetic environment. The resulting circuits, named Josephson parametric converters (JPC), are discussed in detail, and experimental data on their dynamic range are found to be in good agreement with theoretical predictions. We also discuss future prospects and requisite optimization of JPC as a preamplifier for qubit readout applications.
pacs:84.30.Le, 85.25.Cp, 85.25.-j, 42.60.Da
The photon energy of microwave radiation in the band from GHz ( cm wavelength) is approximately smaller than that of the visible light. Yet, at a temperature smaller than room temperature, now routinely achievable with a dilution refrigerator, it is now possible to resolve the energy of single microwave photons (1). There are three advantages of single photon microwave electronics when compared with quantum optics. First, signal shapes at carrier frequencies of a few GHz with a relative bandwidth of few percent can be controlled with much greater relative precision than their equivalent at a few hundreds of THz. This is partly due to the fact that microwave generators have more short term stability than lasers, but also because microwave components are mechanically very stable, particularly when cooled, compared with traditional optical components. Second, in single photon microwave electronics, the on-chip circuitry can be well in the lumped element regime, and spatial mode structure can be controlled more thoroughly and more reliably than in the optical domain. Finally, there exists a simple, robust non-dissipative component, the Josephson tunnel junction (JJ), whose non-linearity can be ultra-strong even at the single photon level (2). Many quantum signal processing functions have been realized using JJs, both digital and analog, and this short review will not attempt to describe all of them. We will focus on analog Josephson devices pumped with a microwave tone. They recently led to microwave amplifiers working at the single photon level (3); (4). These novel devices have taken the work pioneered by B. Yurke at Bell labs 25 years ago (5); (6); (7) to the point where actual experiments can be performed using Josephson amplifiers as the first link in the chain of measurement (8); (9); (10).
In this paper, we address one particular subclass of analog signal processing devices based on Josephson tunnel junction, namely those performing non-degenerate three-wave mixing. Examples are Josephson circuits based on the Josephson ring modulator (11); (12) which we will describe below. The Hamiltonian of such a device is of the form
where (, , ) and (, , ) are the generalized position and momentum variables for the three independent oscillators, and represent the “mass” and “spring constant” of the relevant oscillator (see table I), and is the three-wave mixing constant which governs the non-linearity of the system. We will discuss later how such simple minimal non-linear term can arise. The classical equation of motions for the standing waves in such a device are symmetric and are given by:
where (we assume, for simplicity, equal masses ) and are the angular resonant frequencies of the three coordinates satisfying
We also suppose the oscillators are well in the underdamped regime
a sufficient but not strictly necessary hypothesis, which has the principal merit of keeping the problem analytically soluble under the conditions of interest. It is worth noting that the system is non-degenerate both spatially and temporally. On the other hand, it is important to suppose that the envelope functions , and of the drive signals are supposed to be slow compared to the respective drive frequencies .
The equations (2-4) must be contrasted with that of a degenerate three-wave mixing device for which two cases are possible. In the first case, where the and degrees of freedom have merged into a single oscillator, the Hamiltonian has a non-linear term of the form and the equations read:
This is the case of electromechanical resonators (13) in which one of the capacitance plates of a microwave oscillator () is itself the mass of a mechanical resonator (). There , and pumping the microwave oscillator in the vicinity of leads to cooling of the mechanical oscillator provided . In the second case, it is the and the degrees of freedom that merge into a single oscillator, leading to a non-linear term in the Hamiltonian of the form . The equations then read
and we have now
This case is implemented in Josephson circuits as a dcSQUID whose flux is driven by a microwave oscillating signal at twice the plasma frequency of the SQUID (14). When (so-called “stiff” or “non-depleted” pump condition), the system of equations (11,12) reduces to the parametrically driven oscillator equation
Note that there is, in addition to the parametric drive on the left hand side, a small perturbing drive signal on the right hand side. The theory of the degenerate parametric amplifier starts with this latter equation, the term corresponding to the pump and corresponding to the input signal. The output signal is obtained from a combination of the loss term and the input signal.
In the context of Josephson devices, another route to the effective parametric oscillator of equation (14) can be obtained by a driven, Duffing-type oscillator (15); (16). This system (Josephson bifurcation amplifier) has only one spatial mode and quartic non-linearity,
Driven by a strong tone in the vicinity of the bifurcation occurring at
it will lead to an equation of the form (14) for small deviations around the steady-state solution. It will, therefore, amplify the small drive modulation signal of equation (15) [(17)]. Similar amplifying effects can be found in pumped superconducting microwave resonators without Josephson junctions (18); (19); (20).
In the following section, we will treat Eqs. (2-4) using input-output theory (21) and obtain the quantum-mechanical scattering matrix of the signal and idler amplitudes in the stiff-pump approximation. This allows us to find the photon gain of the device in its photon amplifier mode as a function of the pump amplitude, and the corresponding reduction of bandwidth. We then discuss the implementation of the device using a ring of four Josephson junctions flux-biased at half-quantum in Sec. II. It is the non-dissipative analogue of the semiconductor diode ring modulator (22). In Sec. III, we treat the finite amplitude of signals and establish useful relations between the dynamic range, gain and bandwidth. In Sec. IV we introduce the Josephson parametric converter (JPC) as an example of a non-degenerate, three-wave mixing device operating at the quantum limit. We present three different realizations schemes for the JPC and point out their practical advantages and limitations. In Sec. V we present experimental results for different JPC devices and compare the data with the maximum bounds predicted by theory. We follow this with a discussion, in Sec. VI, of general requirements for an amplifier to meet the needs of qubit readout and how the maximum input power of the device can be increased by two orders of magnitude beyond typical values achieved nowadays. We conclude with a brief summary of our results in Sec. VII.
I Input-output treatment of a generic non-degenerate, three-wave mixing device
The three oscillators of Eqs. (2-4) correspond to three quantum LC oscillators coupled by a non-linear, trilinear mutual inductance, whose mechanism we will discuss in the next section. They are fed by transmission lines which carry excitations both into and out of the oscillators, as shown on Fig. 1. The Hamiltonian of the system is (leaving out the transmission lines for the moment),
where , and are the annihilation operators associated with each of the three degrees of freedom. Their associated angular frequencies are given in terms of the inductances and capacitances as
The bosonic operators of different modes (a, b, c) commute with each other and those associated with the same mode satisfy the usual commutation relations of the form
The link between the mode amplitude such as , which represents the flux through the inductance of the oscillator, and a quantum operator such as can be written as,
where “ZPF” stands for “zero-point fluctuations” and
the last equation defining the impedance of the oscillator, equal to the modulus of the impedance on resonance of either the inductance or the capacitance. The link between and is therefore
We now work in the framework of Rotating Wave Approximation (RWA), in which we only keep terms commuting with the total photon number
Treating in RWA the coupling of each oscillator with a transmission line carrying waves in and out of the oscillator (see Appendix for complements of the next 6 equations), one arrives at three coupled quantum Langevin equations for , and :
In these equations, the second term in the right hand side corresponds to the non-linear term producing photon conversion. The third term says that photons introduced in one resonator leave with a rate
with the resistances denoting the characteristic impedances of the transmission lines. Finally, in the fourth term of the Langevin equations, the input fields such as correspond to the negative frequency component of the drive terms in the classical equations. They obey the relation
where are the usual field operators obeying the commutation relations
in which denotes a frequency that can be either positive or negative. The transmission lines thus both damp and drive the oscillators. The incoming field operator treats the drive signals and the Nyquist equilibrium noise of the reservoir on the same footing. Photon spectral densities of the incoming fields, introduced by relations of the form
have the value
where is the photon flux of the incoming drive signal at angular frequency (in units of photons per unit time) and is the temperature of the electromagnetic excitations of the line. Note that the dimensionless function is defined for both positive and negative frequencies. It is symmetric and its value at frequency represents the average number of photons per unit time per unit bandwidth in the incoming signal, which in the high temperature limit is . It includes the contribution of zero-point quantum noise.
It is worth insisting that we treat the non-linear coupling strength as a perturbation compared with the influence of the reservoirs, treated themselves as a perturbation compared with the Hamiltonian of the oscillators:
In general, only one strong drive tone is applied to one of the resonators and is called the “pump”. Two cases must then be distinguished at this stage, as shown in Fig. 2:
Case 1 (amplification and frequency conversion with photon gain): the pump tone is applied to the resonator. The device is usually used as an amplifier (4); (12). It can also be used as a two-mode squeezer (23).
Case 2 (noiseless frequency conversion without photon gain): the pump tone is applied to either the or resonator (24). The device is useful as a noiseless up- and down-converter and can perform dynamical cooling of the lowest energy oscillator, transferring its spurious excitations to the highest frequency one, which is more easily void of any excitations and plays the role of a cold source.
i.1 Photon gain (case 1)
We will first suppose that the pump is “stiff”, namely
This means that the pump tone will not be easily depleted despite the fact that its photons are converted into the signal and idler photons at and . For solving the quantum Langevin equations, we replace the pumped oscillator annihilation operator by its average value in the coherent state produced by the pump as
The Langevin equations can then be transformed into the linear equations (see equation (178) of Appendix)
After a Fourier transform, we obtain in the frequency domain, a simpler relation
and the signal and idler angular frequencies and are both positive, satisfying the relationship
The scattering matrix of the device for small signals is defined by
It can be computed from Eq. (40) and one finds
where the ’s are the bare response functions of modes a and b (whose inverses depend linearly on the signal frequency)
and is the dimensionless pump amplitude
Note that the matrix in Eq. (43) has unity determinant and the property
For zero frequency detuning, i.e. , the scattering matrix displays a very simple form
where . The zero frequency detuning power gain is given by
For non-zero detuning, the scattering matrix acquires extra phase factors but the minimal scattering matrix for a quantum-limited phase-preserving amplifier represented in Fig. 3 still describes the device.
The gain diverges as ,
i.e. when the photon number in the pump resonator reaches the
critical number given by
a result that is common to all forms of parametric amplification. Increasing the pump power beyond the critical power yielding leads to the parametric oscillation regime. This phenomenon is beyond the scope of our simple analysis and cannot be described by our starting equations, since higher order non-linearities of the system need to be precisely modelled if the saturation of the oscillation is to be accounted for.
Introducing the detuning
we can give a useful expression for the gain as a function of frequency as
which shows that in the limit of large gain, the response of the amplifier for both the signal and idler port is Lorentzian with a bandwidth given by
The product of the maximal amplitude gain times the bandwidth is thus constant and is given by the harmonic average of the oscillator bandwidths. Another interesting prediction of the scattering matrix is the two-mode squeezing function of the device demonstrated in Ref. (25).
i.2 Conversion without photon gain (case 2)
The case of conversion without photon gain can be treated along the same line as in the previous subsection, where scattering takes place between c and a or c and b modes. Without loss of generality we assume that the pump is applied to the intermediate frequency resonance. In this case the scattering matrix reads
The reduced pump strength plays the same role here as in the photon amplification case. Note that the scattering matrix is now unitary (conservation of total number of photons) and satisfies the following relations:
For zero frequency detuning, i.e. , the scattering matrix can be written as
which corresponds to replacing the parameter by or by in the scattering matrix (53). A scattering representation of the two-port device in conversion mode is shown in Fig. 4. In this mode the device operates as a beam splitter, the only difference being that the photons in different arms have different frequencies (24). Full conversion is obtained on resonance when the pump power reaches the critical value. However, here, the critical value can be traversed without violating the validity of the equations. Full photon conversion is desirable in dynamical cooling: in that case, the higher frequency resonator will be emptied of photons, and the lower frequency resonator can be cooled to its ground state by pumping the intermediate frequency resonator (see lower panel of Fig. 2).
i.3 Added Noise
The number of output photons generated per mode in the amplification (case 1) is given by
where is the input photon spectral density given by Eq. (31) and we assume that there is no cross-correlations between the input fields and .
Assuming that the three-wave mixing device is in thermal equilibrium at temperature and that the dominant noise entering the system at each port is zero-point fluctuations (), then in the limit of high gain , the number of noise equivalent photons effectively feeding the system is
This means that the number of noise equivalent photons added by the device to the input is given by . Hence, when operated as a non-degenerate amplifier with , the device adds noise which is equivalent to at least half a photon at the signal frequency to the input, in agreement with Caves theorem (26).
In contrast, in the conversion mode of operation, assuming that there is no correlation between the input fields, the number of generated output photons per mode reads
Therefore, in pure conversion where and , when referring the noise back to the input, one gets noise equivalent photons
This means that, as a converter, the device is not required to add noise to the input since .
Ii Three-wave mixing using JRM
The Josephson ring modulator is a device consisting of four Josephson junctions, each with critical current forming a ring threaded by a flux where is the flux quantum (see Fig. 5). The device has the symmetry of a Wheatstone bridge.
There are thus three orthogonal electrical modes coupled to the junctions, corresponding to the currents , and flowing in three external inductances , and that are much larger than the junction inductance , where is the reduced flux quantum. Each junction is traversed by a current and at the working point (i.e. ) its energy is, keeping terms up to order four in , given by
where and . The currents in the junctions are expressed by
where is the supercurrent induced in the ring by the externally applied flux . The total energy of the ring is, keeping terms up to third order in the currents (27),
We can express the currents as
where are the mutual inductances between and the oscillator inductances . The non-linear coefficient in the energy is, therefore,
and we finally arrive at the result
Here the participation ratios are defined as
and, at ,
The participation ratios are linked to the maximal number of photons in each resonator, defined as those corresponding to an oscillation amplitude reaching a current of in each junction of the ring modulator,
where the are of order with factors accounting for the different participation of modes and in the current of each junction. Equations (78) and (81) are valid for all types of coupling between the Josephson ring modulator and signal/pump oscillators, which can be realized in practice by inductance sharing rather than by the mutual inductances discussed here.
Equation (81) can also be rewritten in terms of the maximum circulating power in cavities a and b as
where we substituted as an upper bound for . The maximum number of photons in equation (81) determine the maximum signal input power handled by the device
We can now combine the notion of maximum power in resonator compatible with weak non-linearity with that of a critical power for the onset of parametric oscillation given by Eq. (55):
arriving at the important relation
where is a number of order unity depending on the exact implementation of the coupling between the ring modulator and the oscillators. The quality factors of the resonators obey the well-known relation
Another maximum limit on the gain of the amplifier is set by the saturation of the device due to amplified zero-point fluctuations present at the input given by
|1 - 16 GHz|
|50 - 500|
|10 - 150|
|0.5 - 10 GHz|
|0.5 - 10 A|
|10 - 230 K|
|0.01 - 0.5|
|0.1 - 15 MHz|
In table II we enlist general bounds on the characteristic parameters of the three-wave mixing device, which are feasible with superconducting microwave circuits and standard Al-AlOx-Al junction fabrication technology. A few comments regarding the values listed in the table are in order. The frequency ranges of resonators a and b is mainly set by the center frequency of the system whose signal one needs to amplify or process. It is also important that these frequencies are very small compared to the plasma frequency of the Josephson junction. The total quality factor range listed in the table is suitable for practical devices. Quality factors in excess of can be easily achieved with superconducting resonators but, as seen from Eq. (58), higher the quality factor, smaller the dynamical bandwidth of the device. Quality factors lower than on the other hand are not recommended either for a variety of reasons. For example, in the limit of very low the pump softens (becomes less stiff), and the dynamic range decreases as more quantum noise will be admitted by the device bandwidth and amplified “unintentionally” by the junctions. The characteristic impedance of the resonators is set by microwave engineering considerations as discussed in Sec. IV but, in general, this value varies around 50 . The rate at which pump photons leave the circuit varies from one circuit design to the other as discussed in Sec. IV and is limited by . This parameter also affects the maximum input power performance of the device as explained in Sec. III. As to the values of , on the one hand it is beneficial to work with large Josephson junctions in order to increase the processing capability of the device; on the other hand a critical current larger than 10 A adds complexity to the microwave design of the resonators and makes the fabrication process of the Josephson junction more involved. This might even require switching to a different fabrication process such as Nb-AlOx-Nb trilayer junctions (28) or nanobridges (29). The other parameters listed in the table, namely , , , , their values depend, to a large extent, on the device parameters already discussed.
Iii Limitation of dynamic range due to pump depletion
In the last two sections, we were using results obtained by solving only the first two of the equations of motion Eqs. (26) under the restriction of the stiff pump approximation. In this section, we extend our analysis and include the third equation describing the dynamics of the pump to calculate the pump depletion and its effect on the dynamic range of the device. For this purpose, we consider the average value of the third equation of motion for field
In steady state and using RWA we obtain
In the limit of vanishing input, the cross-correlation term is negligible and, therefore,
The average number of photons in the resonator in this case is, thus,
We now establish a self-consistent equation for , taking into account input signals of finite amplitude. We first evaluate the value of in the frame rotating with the pump phase,
Using the field relations (see Appendix)
and the input-output relations given by Eq. (43), we obtain (transforming back into the time domain)
where, in the limit of large gains ,
denotes an effective decay rate of pump photons due to generation of entangled signal and idler photons. This last relation expresses, in another form, the Manley-Rowe relations (30) that establish the equality between the number of created signal photons by the amplifier to the number of destroyed pump photons. It shows that even in the absence of any deterministic signal applied to the oscillator or , pump photons are used to amplify zero-point fluctuations. Therefore, the pump tone always encounters a dissipative load even when no signals are injected into the device.
For a continuous wave (CW) input power sent at the center frequency of the or oscillator, or both, we have
where is given in units of photon number per unit time and, in steady state,
As a finite input power is applied to the signal oscillators, oscillator depopulates and, keeping the pump power constant, we get
where denotes the gain in the presence of . In the large gain limit, if we fix the maximum decrease of gain due to pump depletion to be
with , then we obtain
which can also be rewritten as
This relation shows that the ratio of the power of the pump tone to that of the signal at the output of the amplifier must always be much larger than the amplitude gain, in order for the linearity of the amplifier not to be compromised by pump depletion effects.
In Fig. 6 we plot a calculated response of the signal output power versus the signal input power for a typical three-wave mixing device. The device parameters employed in the calculation and listed in the figure caption are practical values yielding a maximum input power, which is limited by the effect of pump depletion. The different blue curves are obtained by solving Eq. (98) for and using the input-output relation , where expressed in units of power is taken as the independent variable and is treated as a parameter. Note that in solving Eq. (98), equations (97), (91), (50) and (54) are used. When drawn on logarithmic scale, the device gain translates into a vertical offset (arrow indicating ) off the line, indicated in red. The dashed black vertical line corresponds to a signal input power of 1 photon at the signal frequency per inverse dynamical bandwidth of the device at dB. The dashed green line corresponds to the maximum gain set by the amplified zero-point fluctuations given by Eq. (87), while the cyan line corresponds to the maximum circulating power in the cavity given by Eq. (82).